Pre-Calculus: Functions

Okay, here is a comprehensive pre-calculus lesson on functions, designed with depth, clarity, and engagement in mind. This is a substantial lesson, aiming to provide a complete learning experience.

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## 1. INTRODUCTION

### 1.1 Hook & Context

Imagine you're designing a self-driving car. One of the core components is the car's ability to understand its surroundings and make decisions based on that information. For example, the car needs to know how far away an object is (a pedestrian, another car, a traffic light) and how fast it's approaching. This information, often gathered by sensors, is then used to calculate braking distance, steering adjustments, and acceleration. At the heart of all these calculations are functions. A function takes an input (like distance) and produces an output (like braking force needed).

Think about your smartphone. When you type a search query into Google, the search engine uses a complex function to analyze your query and return the most relevant results. Or consider a video game where the position of a character is constantly updated based on the player's input (joystick movements, button presses). Again, functions are the engine driving these dynamic changes. Understanding functions is not just about solving equations; it's about understanding how things relate to each other in a quantifiable way, and how those relationships can be used to solve real-world problems.

### 1.2 Why This Matters

The concept of a function is foundational to all advanced mathematics, including calculus, linear algebra, differential equations, and beyond. Without a solid grasp of functions, these more advanced topics become significantly more challenging. Functions are not just abstract mathematical objects; they are powerful tools for modeling real-world phenomena. From predicting weather patterns to designing efficient algorithms, functions are used extensively in science, engineering, economics, and computer science.

Furthermore, a deep understanding of functions fosters critical thinking skills such as problem-solving, logical reasoning, and analytical thinking. These skills are highly valued in a wide range of careers. For example, data scientists use functions to analyze data and build predictive models, engineers use functions to design and optimize systems, and economists use functions to model economic behavior. This knowledge builds directly upon algebra and geometry, extending the ability to describe relationships mathematically. This understanding will later be crucial for calculus, where you'll be analyzing rates of change of functions, and statistics, where you'll be working with probability distributions (which are functions).

### 1.3 Learning Journey Preview

In this lesson, we will embark on a journey to explore the world of functions. We will begin by defining what a function is and how it differs from other mathematical relationships. We will then delve into different ways to represent functions, including equations, graphs, tables, and verbal descriptions. We will learn how to evaluate functions, determine their domain and range, and analyze their key characteristics, such as intercepts, symmetry, and intervals of increase and decrease. Finally, we will explore various types of functions, including linear, quadratic, polynomial, rational, exponential, and logarithmic functions, and discuss their applications in real-world scenarios. Each concept will build on the previous one, providing you with a comprehensive understanding of functions.

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## 2. LEARNING OBJECTIVES

By the end of this lesson, you will be able to:

Explain the definition of a function and differentiate between functions and relations using the vertical line test.
Represent functions using equations, graphs, tables, and verbal descriptions, and translate between these different representations.
Evaluate functions for given inputs, including algebraic expressions and real numbers, and interpret the meaning of the output in context.
Determine the domain and range of various types of functions, including linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
Analyze the key characteristics of functions, such as intercepts, symmetry (even, odd, neither), intervals of increase and decrease, and end behavior.
Identify and apply transformations (shifts, stretches, reflections) to the graphs of functions and write the corresponding equations.
Perform algebraic operations on functions (addition, subtraction, multiplication, division, composition) and determine the domain of the resulting function.
Model real-world scenarios using functions and interpret the meaning of the function in the context of the problem.

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## 3. PREREQUISITE KNOWLEDGE

Before diving into functions, it's crucial to have a solid foundation in the following areas:

Algebraic Expressions: Understanding how to manipulate and simplify algebraic expressions, including variables, constants, and exponents.
Solving Equations: Being able to solve linear, quadratic, and simple polynomial equations.
Graphing: Familiarity with the Cartesian coordinate system and the ability to plot points and graph linear equations.
Inequalities: Understanding how to solve and graph inequalities.
Basic Set Theory: Knowledge of sets, set notation, and set operations (union, intersection).
Interval Notation: Representing sets of numbers using interval notation (e.g., [a, b], (a, ∞)).

Foundational Terminology:

Variable: A symbol (usually a letter) that represents a quantity that can vary.
Constant: A fixed value.
Equation: A statement that two expressions are equal.
Graph: A visual representation of a relationship between two or more variables.
Coordinate Plane: A two-dimensional plane formed by two perpendicular number lines (the x-axis and the y-axis).

If you need to review any of these concepts, consult your algebra textbook or online resources like Khan Academy. A strong grasp of these basics will make learning about functions much smoother.

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## 4. MAIN CONTENT

### 4.1 What is a Function?

Overview: A function is a fundamental concept in mathematics that describes a specific relationship between two sets of elements. It's a rule that assigns each element from one set (the input) to exactly one element in another set (the output).

The Core Concept: Imagine a vending machine. You put in money (the input), and it gives you a specific snack (the output). You can't put in the same amount of money and get two different snacks – the machine is designed to provide a unique output for each input. This is essentially how a function works.

Formally, a function is a relation between a set of inputs called the domain and a set of possible outputs called the range with the property that each input in the domain is related to exactly one output in the range. Think of the domain as the set of all allowable inputs, and the range as the set of all possible outputs that the function can produce. The key is the "exactly one output" condition. If an input can produce multiple outputs, it's not a function, it's just a relation.

We often use the notation f(x) to represent a function, where f is the name of the function and x is the input. f(x) represents the output of the function when the input is x. For example, if f(x) = x², then f(2) = 2² = 4. This means that when the input is 2, the output is 4.

It's important to distinguish between a function and a relation. A relation is simply a set of ordered pairs. A function is a special type of relation that satisfies the "exactly one output" condition. All functions are relations, but not all relations are functions.

Concrete Examples:

Example 1: The Square Function
Setup: Consider the function f(x) = x². This function takes any real number as input and squares it.
Process:
1. Choose an input, say x = 3.
2. Substitute x = 3 into the function: f(3) = 3² = 9.
3. The output is f(3) = 9.
Result: The function f(x) = x² maps the input 3 to the output 9.
Why this matters: This demonstrates the fundamental operation of a function: taking an input and producing a unique output based on a defined rule.

Example 2: The Temperature Conversion Function
Setup: The function C(F) = (5/9)(F - 32) converts Fahrenheit temperature (F) to Celsius temperature (C).
Process:
1. Choose a Fahrenheit temperature, say F = 68°F.
2. Substitute F = 68 into the function: C(68) = (5/9)(68 - 32) = (5/9)(36) = 20.
3. The output is C(68) = 20°C.
Result: The function C(F) converts 68°F to 20°C.
Why this matters: This shows how functions can model real-world relationships and perform useful conversions.

Analogies & Mental Models:

Think of it like... a machine that takes raw materials (input) and transforms them into a finished product (output). The machine follows a specific set of instructions (the function rule) to ensure that the same raw materials always produce the same finished product.
Explain how the analogy maps to the concept: The raw materials are the input (x), the machine is the function (f), and the finished product is the output (f(x)). The machine's instructions are the function's rule.
Where the analogy breaks down (limitations): A machine can break down or have inconsistent outputs. A mathematical function, by definition, always produces the same output for the same input.

Common Misconceptions:

❌ Students often think that f(x) means f multiplied by x.
✓ Actually, f(x) represents the output of the function f when the input is x. It's a notation for indicating the output value, not a multiplication.
Why this confusion happens: The parentheses can be misleading, as they are often used to indicate multiplication.

Visual Description:

Imagine a diagram with two circles. One circle represents the domain (set of inputs), and the other circle represents the range (set of possible outputs). Arrows connect elements in the domain to elements in the range. The key visual element is that each element in the domain has exactly one arrow leaving it, pointing to an element in the range. If any element in the domain has more than one arrow leaving it, it's not a function.

Practice Check:

Is the following relation a function? {(1, 2), (2, 4), (3, 6), (1, 8)}

Answer with explanation: No, this is not a function. The input 1 is associated with two different outputs, 2 and 8, which violates the "exactly one output" condition.

Connection to Other Sections:

This section lays the foundation for all subsequent sections. Understanding the definition of a function is crucial for understanding how to represent, evaluate, analyze, and apply functions. This leads directly to the next section on different ways to represent functions.

### 4.2 Representing Functions

Overview: Functions can be represented in various ways, each providing a different perspective on the relationship between inputs and outputs. These representations include equations, graphs, tables, and verbal descriptions.

The Core Concept: Representing a function is like describing the same object in different languages. Each representation offers a unique way to understand the function's behavior and properties.

Equations: An equation is a mathematical expression that defines the relationship between the input (usually x) and the output (usually y or f(x)). For example, y = 2x + 1 is an equation representing a linear function.
Graphs: A graph is a visual representation of a function on a coordinate plane. The x-axis represents the input, and the y-axis represents the output. Each point on the graph corresponds to an ordered pair (x, y) that satisfies the function's equation.
Tables: A table is a list of ordered pairs (x, y) that shows the relationship between inputs and outputs. Tables are useful for representing functions with a finite number of inputs or when the function's equation is unknown.
Verbal Descriptions: A verbal description is a written explanation of how the function works. For example, "the function doubles the input and adds 3" describes the function f(x) = 2x + 3.

Being able to translate between these different representations is a crucial skill in pre-calculus. For example, you should be able to take an equation and create a graph, or take a table of values and write an equation that represents the relationship.

Concrete Examples:

Example 1: Linear Function
Equation: y = -x + 5
Graph: A straight line with a slope of -1 and a y-intercept of 5.
Table:

| x | y |
|-----|-----|
| -2 | 7 |
| 0 | 5 |
| 2 | 3 |
| 4 | 1 |
Verbal Description: "The function takes an input, multiplies it by -1, and then adds 5."
Why this matters: This demonstrates how the same linear relationship can be expressed in four different ways.

Example 2: Quadratic Function
Equation: f(x) = x² - 4x + 3
Graph: A parabola opening upwards, with vertex at (2, -1) and x-intercepts at 1 and 3.
Table:

| x | f(x) |
|-----|------|
| 0 | 3 |
| 1 | 0 |
| 2 | -1 |
| 3 | 0 |
| 4 | 3 |
Verbal Description: "The function takes an input, squares it, subtracts 4 times the input, and then adds 3."
Why this matters: This showcases the different representations for a non-linear function.

Analogies & Mental Models:

Think of it like... a multifaceted jewel. Each facet (equation, graph, table, verbal description) reflects the light (the function) in a different way, revealing different aspects of its beauty and structure.
Explain how the analogy maps to the concept: Each representation highlights different properties of the function, just as each facet of a jewel reflects light differently.
Where the analogy breaks down (limitations): A jewel is a static object, while a function represents a dynamic relationship.

Common Misconceptions:

❌ Students often think that a table is only a valid representation of a function if it includes all possible inputs.
✓ Actually, a table can represent a function even if it only includes a subset of the domain. The key is that each input in the table must be associated with exactly one output.
Why this confusion happens: Students may associate functions exclusively with equations and graphs, where the entire domain is typically represented.

Visual Description:

Imagine four panels displayed side-by-side. The first panel shows an equation (e.g., y = mx + b). The second panel shows a graph (a line on a coordinate plane). The third panel shows a table of x and y values. The fourth panel shows a written description of the function's rule. Key visual elements are the connection between these panels – how the same relationship is expressed in different visual and symbolic forms.

Practice Check:

Write a verbal description for the function represented by the equation f(x) = √(x + 2).

Answer with explanation: "The function takes an input, adds 2 to it, and then takes the square root of the result."

Connection to Other Sections:

This section builds on the definition of a function and prepares you for evaluating and analyzing functions. Being able to represent functions in different ways is essential for understanding their properties and behavior, which will be covered in subsequent sections. This leads to the next section on evaluating functions.

### 4.3 Evaluating Functions

Overview: Evaluating a function means finding the output value (f(x)) for a given input value (x). This is a fundamental skill for working with functions and understanding their behavior.

The Core Concept: Function evaluation is like using the function as a machine. You feed in an input, and the machine processes it according to the function's rule, producing an output.

To evaluate a function, you simply substitute the given input value for the variable in the function's equation and simplify. For example, if f(x) = 3x - 2 and you want to find f(4), you substitute x = 4 into the equation: f(4) = 3(4) - 2 = 12 - 2 = 10. Therefore, f(4) = 10.

Function evaluation can also involve algebraic expressions as inputs. For example, if f(x) = x² + 1 and you want to find f(a + 1), you substitute (a + 1) for x in the equation: f(a + 1) = (a + 1)² + 1 = a² + 2a + 1 + 1 = a² + 2a + 2.

Concrete Examples:

Example 1: Evaluating a Polynomial Function
Setup: Consider the function f(x) = x³ - 2x + 1. Evaluate f(-2).
Process:
1. Substitute x = -2 into the function: f(-2) = (-2)³ - 2(-2) + 1.
2. Simplify: f(-2) = -8 + 4 + 1 = -3.
Result: f(-2) = -3.
Why this matters: This demonstrates how to evaluate a function with a negative input.

Example 2: Evaluating with an Algebraic Expression
Setup: Consider the function g(x) = 2x² - x. Evaluate g(x + h).
Process:
1. Substitute x + h for x in the function: g(x + h) = 2(x + h)² - (x + h).
2. Expand and simplify: g(x + h) = 2(x² + 2xh + h²) - x - h = 2x² + 4xh + 2h² - x - h.
Result: g(x + h) = 2x² + 4xh + 2h² - x - h.
Why this matters: This shows how to evaluate a function with an algebraic expression as input, which is essential for understanding concepts like the difference quotient in calculus.

Analogies & Mental Models:

Think of it like... a calculator. You enter a number (the input), press a button (the function), and the calculator displays the result (the output).
Explain how the analogy maps to the concept: The calculator performs a specific operation (the function) on the input to produce the output.
Where the analogy breaks down (limitations): A calculator has a limited set of functions, while mathematical functions can be defined with infinite complexity.

Common Misconceptions:

❌ Students often make mistakes when evaluating functions with negative inputs, forgetting to apply the exponent correctly.
✓ Actually, pay close attention to the order of operations and the signs of the numbers. For example, (-2)² = 4, not -4.
Why this confusion happens: Carelessness with negative signs is a common error in algebra.

Visual Description:

Imagine a function machine with an input slot and an output slot. You put a number into the input slot, and the machine processes it according to the function's rule. The resulting number comes out of the output slot. Key visual elements are the input going into the machine, the function's rule being applied inside the machine, and the output emerging from the machine.

Practice Check:

If h(x) = (x - 1) / (x + 1), find h(0).

Answer with explanation: h(0) = (0 - 1) / (0 + 1) = -1 / 1 = -1.

Connection to Other Sections:

This section builds on the representation of functions and prepares you for determining the domain and range of functions. Evaluating functions is essential for understanding their behavior and graphing them, which will be covered in subsequent sections. This leads to the next section on domain and range.

### 4.4 Domain and Range

Overview: The domain and range are fundamental concepts that describe the set of possible inputs and outputs for a function. Understanding the domain and range is crucial for interpreting the function's behavior and its applicability to real-world scenarios.

The Core Concept: The domain is like the allowed "ingredients" for a recipe (the function), and the range is the set of all possible "dishes" (outputs) that can be created using those ingredients.

The domain of a function is the set of all possible input values (x) for which the function is defined. In other words, it's the set of all values of x that you can plug into the function without causing any mathematical errors, such as division by zero or taking the square root of a negative number.

The range of a function is the set of all possible output values (f(x)) that the function can produce. In other words, it's the set of all values that f(x) can take on as x varies over the domain.

Determining the domain and range of a function often involves analyzing the function's equation and considering any restrictions that might apply. For example:

Rational Functions: The domain excludes any values of x that make the denominator equal to zero.
Radical Functions (even root): The domain excludes any values of x that make the expression under the radical negative.
Logarithmic Functions: The domain excludes any values of x that are zero or negative.

Concrete Examples:

Example 1: Finding the Domain and Range of a Linear Function
Setup: Consider the function f(x) = 2x + 3.
Process:
1. Domain: Since there are no restrictions on the input values (no division by zero, no square roots of negative numbers), the domain is all real numbers.
2. Range: Since the function is a linear function with a non-zero slope, it can take on any real number as an output. Therefore, the range is all real numbers.
Result:
Domain: (-∞, ∞)
Range: (-∞, ∞)
Why this matters: This shows that linear functions (with non-zero slope) are defined for all real numbers and can produce any real number as output.

Example 2: Finding the Domain and Range of a Rational Function
Setup: Consider the function g(x) = 1 / (x - 2).
Process:
1. Domain: The function is undefined when the denominator is zero, which occurs when x = 2. Therefore, the domain is all real numbers except 2.
2. Range: The function can take on any real number as output except for 0. This is because the numerator is always 1, so the fraction can never be equal to zero.
Result:
Domain: (-∞, 2) ∪ (2, ∞)
Range: (-∞, 0) ∪ (0, ∞)
Why this matters: This demonstrates how to identify and exclude values from the domain and range of a rational function due to division by zero.

Analogies & Mental Models:

Think of it like... a parking lot. The domain is the set of all parking spaces that are available, and the range is the set of all cars that are parked in those spaces.
Explain how the analogy maps to the concept: The parking spaces represent the allowed inputs, and the cars represent the resulting outputs.
Where the analogy breaks down (limitations): A parking lot has a finite number of spaces, while the domain and range of a function can be infinite.

Common Misconceptions:

❌ Students often forget to consider restrictions on the domain when dealing with radical functions, especially even roots.
✓ Actually, remember that the expression under an even root must be non-negative.
Why this confusion happens: Students may focus on the algebraic manipulation of the function and overlook the underlying restrictions on the input values.

Visual Description:

Imagine a graph of a function on a coordinate plane. The domain is the set of all x-values for which the graph exists. You can visualize the domain by projecting the graph onto the x-axis. The range is the set of all y-values for which the graph exists. You can visualize the range by projecting the graph onto the y-axis. The resulting intervals on the x and y axes represent the domain and range, respectively.

Practice Check:

Find the domain of the function h(x) = √(x + 3).

Answer with explanation: The expression under the square root must be non-negative, so x + 3 ≥ 0. Solving for x, we get x ≥ -3. Therefore, the domain is [-3, ∞).

Connection to Other Sections:

This section builds on the evaluation of functions and prepares you for analyzing the key characteristics of functions. Understanding the domain and range is essential for interpreting the function's behavior and its applicability to real-world scenarios. This leads to the next section on key characteristics.

### 4.5 Key Characteristics of Functions

Overview: Analyzing the key characteristics of a function provides a deeper understanding of its behavior and properties. These characteristics include intercepts, symmetry (even, odd, or neither), intervals of increase and decrease, and end behavior.

The Core Concept: Analyzing a function's characteristics is like examining a map to understand the landscape of a function's graph. Each characteristic reveals a different aspect of the function's behavior.

Intercepts:
x-intercepts: The points where the graph intersects the x-axis (where y = 0). To find the x-intercepts, set f(x) = 0 and solve for x.
y-intercepts: The point where the graph intersects the y-axis (where x = 0). To find the y-intercept, evaluate f(0).
Symmetry:
Even Function: A function is even if f(-x) = f(x) for all x in the domain. The graph of an even function is symmetric about the y-axis.
Odd Function: A function is odd if f(-x) = -f(x) for all x in the domain. The graph of an odd function is symmetric about the origin.
Neither: A function that is neither even nor odd.
Intervals of Increase and Decrease:
Increasing: A function is increasing on an interval if its values increase as x increases.
Decreasing: A function is decreasing on an interval if its values decrease as x increases.
Constant: A function is constant on an interval if its values remain the same as x increases.
End Behavior: Describes what happens to the function's values as x approaches positive or negative infinity.

Concrete Examples:

Example 1: Analyzing a Quadratic Function
Setup: Consider the function f(x) = x² - 4x + 3.
Process:
1. Intercepts:
x-intercepts: Set f(x) = 0: x² - 4x + 3 = 0. Factoring, we get (x - 1)(x - 3) = 0. So, x = 1 and x = 3.
y-intercept: f(0) = 0² - 4(0) + 3 = 3.
2. Symmetry: f(-x) = (-x)² - 4(-x) + 3 = x² + 4x + 3. This is neither equal to f(x) nor -f(x), so the function is neither even nor odd.
3. Intervals of Increase and Decrease: The vertex of the parabola is at x = -b / 2a = 4 / 2 = 2. The function is decreasing on the interval (-∞, 2) and increasing on the interval (2, ∞).
4. End Behavior: As x approaches positive or negative infinity, f(x) approaches positive infinity.
Result:
x-intercepts: (1, 0), (3, 0)
y-intercept: (0, 3)
Symmetry: Neither even nor odd
Intervals of Increase and Decrease: Decreasing on (-∞, 2), Increasing on (2, ∞)
End Behavior: As x → ±∞, f(x) → ∞
Why this matters: This demonstrates how to analyze the key characteristics of a quadratic function to understand its shape and behavior.

Example 2: Analyzing a Rational Function
Setup: Consider the function g(x) = 1 / x.
Process:
1. Intercepts:
x-intercepts: There are no x-intercepts because 1 / x can never be equal to zero.
y-intercept: The function is undefined at x = 0, so there is no y-intercept.
2. Symmetry: g(-x) = 1 / (-x) = -1 / x = -g(x). So, the function is odd.
3. Intervals of Increase and Decrease: The function is decreasing on the interval (-∞, 0) and decreasing on the interval (0, ∞).
4. End Behavior: As x approaches positive infinity, g(x) approaches 0. As x approaches negative infinity, g(x) approaches 0.
Result:
x-intercepts: None
y-intercept: None
Symmetry: Odd
Intervals of Increase and Decrease: Decreasing on (-∞, 0), Decreasing on (0, ∞)
End Behavior: As x → ±∞, g(x) → 0
Why this matters: This demonstrates how to analyze the key characteristics of a rational function, including its symmetry and end behavior.

Analogies & Mental Models:

Think of it like... a detective investigating a crime scene. Each characteristic (intercepts, symmetry, intervals of increase/decrease, end behavior) provides a clue about the function's identity and behavior.
Explain how the analogy maps to the concept: The detective gathers clues to understand the crime, just as we analyze the characteristics to understand the function.
Where the analogy breaks down (limitations): A crime scene is unique and complex, while functions follow predictable rules.

Common Misconceptions:

❌ Students often confuse even and odd functions, especially when dealing with more complex expressions.
✓ Actually, remember the definitions: f(-x) = f(x) for even functions and f(-x) = -f(x) for odd functions. Test the function with a specific value of x to help determine its symmetry.
Why this confusion happens: The algebraic manipulation can be challenging, and it's easy to make mistakes with signs.

Visual Description:

Imagine a graph of a function on a coordinate plane. The intercepts are the points where the graph crosses the x and y axes. Symmetry can be visualized by reflecting the graph across the y-axis (for even functions) or rotating it 180 degrees about the origin (for odd functions). Intervals of increase and decrease are the sections of the graph where the y-values are increasing or decreasing as you move from left to right. End behavior describes what happens to the graph as you move infinitely far to the left or right.

Practice Check:

Determine whether the function f(x) = x³ + x is even, odd, or neither.

Answer with explanation: f(-x) = (-x)³ + (-x) = -x³ - x = -(x³ + x) = -f(x). Therefore, the function is odd.

Connection to Other Sections:

This section builds on the domain and range of functions and prepares you for identifying and applying transformations to functions. Understanding the key characteristics of functions is essential for graphing them and modeling real-world scenarios. This leads to the next section on transformations.

### 4.6 Transformations of Functions

Overview: Transformations of functions involve altering the graph of a function by shifting, stretching, reflecting, or compressing it. Understanding transformations allows you to quickly sketch graphs of related functions and analyze their behavior.

The Core Concept: Transformations are like using image editing software to manipulate a photo. You can move it, resize it, flip it, or change its perspective.

The basic transformations include:

Vertical Shifts: Adding a constant c to the function, f(x) + c, shifts the graph vertically by c units. If c > 0, the graph shifts upward. If c < 0, the graph shifts downward.
Horizontal Shifts: Replacing x with (x - c) in the function, f(x - c), shifts the graph horizontally by c units. If c > 0, the graph shifts to the right. If c < 0, the graph shifts to the left.
Vertical Stretches and Compressions: Multiplying the function by a constant a, af(x), stretches the graph vertically if |a| > 1 and compresses the graph vertically if 0 < |a| < 1. If a < 0, the graph is also reflected across the x-axis.
Horizontal Stretches and Compressions: Replacing x with (x / b) in the function, f(x / b), stretches the graph horizontally if |b| > 1 and compresses the graph horizontally if 0 < |b| < 1. If b < 0, the graph is also reflected across the y-axis.
Reflections:
*Reflection

Okay, here is a comprehensive pre-calculus lesson on functions, designed with the requested depth, structure, and detail. This will be a substantial piece of content.

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## 1. INTRODUCTION

### 1.1 Hook & Context

Imagine you're designing a roller coaster. You need to know how high to build each hill, how steep the drops should be, and how fast the coaster will travel at different points. Or perhaps you're creating a new social media app; you need to predict how many users you'll have after a month, a year, or five years based on your initial growth. Both of these scenarios, seemingly worlds apart, rely on the same fundamental mathematical concept: functions. Functions are the mathematical machines that take an input, process it according to a specific rule, and produce a predictable output. Think of a vending machine: you put in money (the input), select a snack (the rule), and get your desired treat (the output).

Maybe you've already encountered functions in algebra. You've seen equations like y = 2x + 3 or f(x) = x². But in pre-calculus, we're going to dive much deeper. We'll explore different types of functions, how they transform, how they combine, and how they model real-world phenomena with incredible accuracy. We'll move beyond simple equations and learn to think about functions as powerful tools for understanding and predicting patterns in the world around us. This isn't just about memorizing formulas; it's about developing a way of thinking that will be invaluable in fields like engineering, computer science, economics, and even the arts.

### 1.2 Why This Matters

Functions are the bedrock of much of advanced mathematics and are indispensable in countless real-world applications. Understanding functions well is not just about getting a good grade in pre-calculus; it's about opening doors to future opportunities. In engineering, functions are used to model everything from the stress on a bridge to the flow of electricity in a circuit. In computer science, functions are the building blocks of programs, allowing us to create complex software applications. Economists use functions to model market trends, predict economic growth, and analyze consumer behavior. Even artists and musicians use functions to generate patterns, create visual effects, and compose music.

This knowledge builds directly on your existing algebra skills, especially your understanding of equations, graphs, and variables. We'll take those foundational concepts and elevate them to a more abstract and powerful level. Looking ahead, a solid grasp of functions is absolutely essential for calculus, differential equations, linear algebra, and many other advanced mathematical topics. Without a strong foundation in functions, these subjects will be significantly more challenging. Furthermore, the analytical and problem-solving skills you develop by studying functions will be valuable in any career path you choose.

### 1.3 Learning Journey Preview

Over the course of this lesson, we will journey through the fascinating world of functions. We'll start by defining exactly what a function is and exploring different ways to represent them (equations, graphs, tables, and verbal descriptions). We'll then delve into the concept of domain and range, understanding the permissible inputs and possible outputs of a function. Next, we'll examine various types of functions, including linear, quadratic, polynomial, rational, exponential, and logarithmic functions, each with its unique properties and behaviors. We'll learn how to transform functions by shifting, stretching, and reflecting their graphs. We'll also explore operations on functions, such as addition, subtraction, multiplication, division, and composition. Finally, we'll apply our knowledge to solve real-world problems, model data, and make predictions. Each concept will build on the previous one, creating a cohesive and comprehensive understanding of functions.

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## 2. LEARNING OBJECTIVES

By the end of this lesson, you will be able to:

1. Define a function and explain the difference between a function and a relation using the vertical line test.
2. Determine the domain and range of a function represented graphically, algebraically, and in table form.
3. Identify and classify different types of functions, including linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
4. Apply transformations (translations, reflections, stretches, and compressions) to functions and describe the resulting changes in their graphs and equations.
5. Perform operations on functions, including addition, subtraction, multiplication, division, and composition, and determine the resulting domain.
6. Model real-world scenarios using functions and interpret the meaning of function values, intercepts, and slopes in context.
7. Analyze the end behavior of polynomial and rational functions and identify horizontal, vertical, and slant asymptotes.
8. Solve application problems involving functions, including optimization problems and exponential growth/decay models.

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## 3. PREREQUISITE KNOWLEDGE

Before embarking on this journey into the world of functions, it’s important to have a solid foundation in the following concepts:

Basic Algebra: You should be comfortable with algebraic manipulations, solving equations (linear, quadratic), and simplifying expressions.
Graphing: Familiarity with the Cartesian coordinate system, plotting points, and graphing linear equations is essential. Review slope-intercept form (y = mx + b).
Inequalities: Understanding how to solve and graph inequalities is crucial for determining the domain and range of functions.
Exponents and Radicals: Knowledge of exponent rules and how to simplify radical expressions is needed for working with exponential and radical functions.
Factoring: Being able to factor polynomials is important for simplifying rational functions and finding roots.
Set Notation: Understanding set notation (e.g., {x | x > 2}, intervals like (-∞, 5]) is key for expressing domains and ranges.

Quick Review: If any of these topics feel rusty, take some time to review them. Khan Academy and other online resources offer excellent refresher courses on algebra and graphing. Make sure you understand the basics before moving on, as they form the building blocks for everything we'll cover.

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## 4. MAIN CONTENT

### 4.1 What is a Function?

Overview: At its core, a function is a specific type of relation between two sets, where each input from the first set (the domain) is associated with exactly one output in the second set (the range). This "one-to-one" (or "many-to-one", but never "one-to-many") relationship is what distinguishes a function from a more general relation. Understanding this fundamental concept is crucial for everything that follows.

The Core Concept: A function is a rule or mapping that takes an element from a set called the domain and assigns it to exactly one element in a set called the range. Think of it as a machine: you put something in (the input), and the machine processes it and spits out something else (the output). The key is that for each input, the machine always produces the same output.

More formally, a function f from a set A to a set B, denoted f: A → B, is a relation that associates each element x in A with a unique element y in B. We write y = f(x), where x is the input, f is the function (the rule), and y is the output. A is the domain of f, and the set of all possible outputs y in B is the range of f. It's important to note that the range is a subset of B. B is often called the codomain.

The crucial condition for a relation to be a function is that for every input x in the domain, there must be only one corresponding output y in the range. If an input x maps to multiple outputs, then it is not a function, only a relation.

Concrete Examples:

Example 1: The Squaring Function
Setup: Consider the function f(x) = x². This function takes any real number x as input and squares it to produce the output y.
Process: If we input x = 3, the function calculates f(3) = 3² = 9. If we input x = -3, the function calculates f(-3) = (-3)² = 9. Notice that two different inputs can produce the same output (this is allowed in a function), but one input cannot produce two different outputs.
Result: The squaring function is a valid function because for every real number x, there is only one possible value of x².
Why this matters: This simple example illustrates the core concept of a function: a well-defined rule that produces a unique output for each input.
Example 2: Not a Function (A Relation)
Setup: Consider the relation x = y². This relates x and y, but is it a function of x?
Process: If we input x = 4, we need to find y such that 4 = y². This equation has two solutions: y = 2 and y = -2.
Result: Since the input x = 4 produces two different outputs, y = 2 and y = -2, this relation is not a function of x. y is a function of x only if the equation can be rearranged into the form y = f(x) where f(x) produces only one output for each x.
Why this matters: This example highlights the importance of the "one-to-one" (or "many-to-one") requirement. A relation can link inputs and outputs in any way, but a function must have a unique output for each input.

Analogies & Mental Models:

Think of it like a machine: You put ingredients (input) into a baking machine. The machine follows a recipe (function) and produces a cake (output). If you put the same ingredients in, you should always get the same cake. If sometimes you get a cake and sometimes you get cookies, it's not a reliable machine, and it doesn't represent a function.
Think of it like a lock and key: The input is the key, and the function is the lock. Each key (input) should open only one lock (output). If a key can open multiple locks, it's not a function.
Where the analogy breaks down: Machines and locks are physical things. Functions are abstract mathematical concepts. The analogy helps visualize the input-output relationship, but it doesn't capture the full complexity of functions.

Common Misconceptions:

❌ Students often think that if a graph is curved, it can't be a function.
✓ Actually, the shape of the graph doesn't determine whether it's a function. The key is whether any vertical line intersects the graph more than once (the vertical line test, explained below).
Why this confusion happens: Students may associate straight lines with linear functions and assume that curves represent something else.

Visual Description:

Imagine a graph in the Cartesian plane. To determine if the graph represents a function, use the Vertical Line Test: Draw a vertical line anywhere on the graph. If the vertical line intersects the graph at more than one point, then the graph does not represent a function. This is because, at the x-value where the vertical line intersects the graph, there are multiple corresponding y-values, violating the "one-to-one" requirement. If every vertical line intersects the graph at most once, then the graph does represent a function.

Practice Check:

Question: Does the equation x² + y² = 1 represent y as a function of x? Why or why not?

Answer: No, it does not. This is the equation of a circle. A vertical line drawn through the circle will intersect it at two points (except at x = -1 and x = 1). Therefore, for a given x-value (between -1 and 1), there are two corresponding y-values, violating the definition of a function.

Connection to Other Sections:

This section lays the foundation for all subsequent sections. Understanding the definition of a function is crucial for determining the domain and range, identifying different types of functions, and performing operations on functions. The vertical line test is a powerful visual tool that will be used throughout the lesson.

### 4.2 Representing Functions

Overview: Functions aren't just abstract mathematical concepts; they can be represented in multiple ways, each offering a different perspective and highlighting different aspects of the function. The most common ways to represent a function are through equations, graphs, tables, and verbal descriptions. Being fluent in translating between these representations is a key skill.

The Core Concept: A function establishes a relationship between input and output. This relationship can be expressed in a variety of formats.
Equation: The most common representation is an equation that defines the rule for calculating the output y (or f(x)) from the input x. For example, y = 3x + 2 or f(x) = x² - 5x + 6.
Graph: A graph is a visual representation of the function in the Cartesian plane. The x-axis represents the input values (domain), and the y-axis represents the output values (range). Each point on the graph corresponds to an ordered pair (x, y) that satisfies the function's equation.
Table: A table lists pairs of input and output values. The table shows a sample of the function's behavior but doesn't necessarily provide a complete picture of the function's behavior.
Verbal Description: A function can also be described in words. For example, "The function doubles the input and then adds 5." This description can then be translated into an equation (f(x) = 2x + 5).

Concrete Examples:

Example 1: Linear Function
Equation: f(x) = 2x - 1
Graph: A straight line with a slope of 2 and a y-intercept of -1.
Table:

| x | f(x) |
| --- | ---- |
| -2 | -5 |
| -1 | -3 |
| 0 | -1 |
| 1 | 1 |
| 2 | 3 |
Verbal Description: "The function multiplies the input by 2 and then subtracts 1."
Example 2: Quadratic Function
Equation: g(x) = x² - 4
Graph: A parabola opening upwards, with its vertex at (0, -4) and x-intercepts at -2 and 2.
Table:

| x | g(x) |
| --- | ---- |
| -3 | 5 |
| -2 | 0 |
| -1 | -3 |
| 0 | -4 |
| 1 | -3 |
| 2 | 0 |
| 3 | 5 |
Verbal Description: "The function squares the input and then subtracts 4."

Analogies & Mental Models:

Think of it like different languages: An equation is like writing a sentence in English, a graph is like drawing a picture, a table is like a list of words, and a verbal description is like telling a story. They all convey the same information but in different formats.
Think of it like different views of a sculpture: An equation is like seeing the sculpture from above, a graph is like seeing it from the front, a table is like seeing it from the side, and a verbal description is like hearing someone describe it.

Common Misconceptions:

❌ Students often think that a table can fully represent a function.
✓ Actually, a table only shows a limited number of input-output pairs. It doesn't necessarily reveal the complete behavior of the function, especially between the listed points.
Why this confusion happens: Students may rely too heavily on the given data points and not consider the function's behavior outside of those points.

Visual Description:

When looking at a graph, pay attention to key features:
x-intercepts: Where the graph crosses the x-axis (where y = 0).
y-intercept: Where the graph crosses the y-axis (where x = 0).
Slope: The steepness of the line (for linear functions).
Vertex: The highest or lowest point on a parabola (for quadratic functions).
Asymptotes: Lines that the graph approaches but never touches (for rational functions).

These visual clues can help you understand the function's behavior and translate it into other representations.

Practice Check:

Question: A function is described verbally as "The function takes the square root of the input and then adds 1." Write the equation, create a table with at least 5 values, and sketch a rough graph of this function.

Answer:
Equation: f(x) = √x + 1
Table:

| x | f(x) |
| --- | --------- |
| 0 | 1 |
| 1 | 2 |
| 4 | 3 |
| 9 | 4 |
| 16 | 5 |
Graph: The graph starts at (0, 1) and increases slowly as x increases. It's a transformation of the basic square root function.

Connection to Other Sections:

This section is essential for understanding how to work with functions in different contexts. Being able to translate between representations allows you to choose the most appropriate representation for a given problem and to gain a deeper understanding of the function's behavior. This skill will be used extensively when analyzing transformations, operations, and applications of functions.

### 4.3 Domain and Range

Overview: The domain and range are fundamental properties of a function that define its scope and limitations. The domain is the set of all possible inputs that the function can accept, while the range is the set of all possible outputs that the function can produce. Understanding these concepts is crucial for interpreting function behavior and solving real-world problems.

The Core Concept:
Domain: The domain of a function f(x) is the set of all possible values of x for which the function is defined. In other words, it's the set of all inputs that will produce a valid output. We need to consider restrictions like:
Division by zero: The denominator of a fraction cannot be zero.
Square roots of negative numbers: The radicand (the expression inside the square root) must be non-negative.
Logarithms of non-positive numbers: The argument of a logarithm (the expression inside the logarithm) must be positive.
Range: The range of a function f(x) is the set of all possible values of f(x) (or y) that the function can produce. In other words, it's the set of all outputs that result from plugging in values from the domain. Determining the range can be more challenging than determining the domain and often requires analyzing the function's graph or equation.

Concrete Examples:

Example 1: Linear Function
f(x) = 3x + 2
Domain: All real numbers (denoted as (-∞, ∞) or ℝ). There are no restrictions on the input x.
Range: All real numbers (-∞, ∞) or ℝ. A linear function with a non-zero slope will produce all possible real number outputs.
Example 2: Rational Function
g(x) = 1/(x - 2)
Domain: All real numbers except x = 2 (denoted as (-∞, 2) ∪ (2, ∞)). The function is undefined when the denominator is zero.
Range: All real numbers except y = 0 (denoted as (-∞, 0) ∪ (0, ∞)). The function can get arbitrarily close to 0 as x gets very large or very small, but it will never actually equal 0.
Example 3: Square Root Function
h(x) = √(x + 3)
Domain: All real numbers greater than or equal to -3 (denoted as [-3, ∞)). The expression inside the square root must be non-negative.
Range: All real numbers greater than or equal to 0 (denoted as [0, ∞)). The square root function always produces non-negative outputs.

Analogies & Mental Models:

Think of the domain as the allowable ingredients for a recipe: You can only use certain ingredients to make a particular dish. The domain is the set of ingredients you're allowed to use.
Think of the range as the possible outcomes of a game: You can only get certain scores in a game. The range is the set of possible scores you can achieve.

Common Misconceptions:

❌ Students often forget to consider restrictions on the domain, especially when dealing with rational or radical functions.
✓ Always check for potential division by zero, square roots of negative numbers, and logarithms of non-positive numbers.
Why this confusion happens: Students may focus solely on the equation and not think about the underlying mathematical principles that govern the function's behavior.

Visual Description:

Domain: On a graph, the domain is the set of all x-values that have a corresponding y-value. Look for gaps, holes, or vertical asymptotes that indicate values excluded from the domain.
Range: On a graph, the range is the set of all y-values that have a corresponding x-value. Look for the highest and lowest points on the graph, as well as horizontal asymptotes, to determine the possible y-values.

Practice Check:

Question: Determine the domain and range of the function f(x) = √(9 - x²).

Answer:
Domain: [-3, 3]. The expression inside the square root must be non-negative, so 9 - x² ≥ 0, which implies x² ≤ 9, and therefore -3 ≤ x ≤ 3.
Range: [0, 3]. The square root function always produces non-negative outputs. The maximum value of √(9 - x²) occurs when x = 0, giving f(0) = √9 = 3.

Connection to Other Sections:

Understanding domain and range is crucial for analyzing transformations of functions. When you shift, stretch, or reflect a function, the domain and range may change accordingly. It's also important for understanding the composition of functions, as the domain of the composite function depends on the domains of the individual functions.

### 4.4 Types of Functions

Overview: Functions come in many forms, each with its own unique characteristics and properties. Understanding the different types of functions is essential for modeling real-world phenomena and solving mathematical problems. We'll cover several key types: linear, quadratic, polynomial, rational, exponential, and logarithmic.

The Core Concept: Each type of function has a specific algebraic form and a characteristic graph. Recognizing these forms and graphs allows you to quickly identify the type of function you're dealing with and apply the appropriate techniques for analyzing it.

Linear Functions: f(x) = mx + b, where m is the slope and b is the y-intercept. Graph: A straight line.
Quadratic Functions: f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. Graph: A parabola.
Polynomial Functions: f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where aₙ, aₙ₋₁, ..., a₁, a₀ are constants and n is a non-negative integer (the degree of the polynomial). Graph: A smooth, continuous curve with at most n - 1 turning points.
Rational Functions: f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials and Q(x) ≠ 0. Graph: Can have vertical and horizontal asymptotes, and may have holes.
Exponential Functions: f(x) = aˣ, where a is a positive constant and a ≠ 1. Graph: Increases or decreases rapidly, has a horizontal asymptote.
Logarithmic Functions: f(x) = logₐ(x), where a is a positive constant and a ≠ 1. Graph: The inverse of an exponential function, has a vertical asymptote.

Concrete Examples:

Example 1: Identifying a Polynomial Function
Function: f(x) = 5x⁴ - 3x² + x - 7
Analysis: This is a polynomial function because it consists of terms with non-negative integer exponents. The degree of the polynomial is 4.
Example 2: Identifying a Rational Function
Function: g(x) = (x + 2)/(x² - 9)
Analysis: This is a rational function because it is a ratio of two polynomials. The denominator can be factored as (x + 3)(x - 3), so the function has vertical asymptotes at x = -3 and x = 3.
Example 3: Identifying an Exponential Function
Function: h(x) = 2^(x+1)
Analysis: This is an exponential function because the variable x appears in the exponent. The base is 2.

Analogies & Mental Models:

Think of each type of function as a different type of animal: A linear function is like a straight-walking ant, a quadratic function is like a jumping dolphin, an exponential function is like a rapidly growing bacteria colony, and a logarithmic function is like the spreading ripples from a stone thrown into a pond.
Think of each type of function as a different tool: A linear function is like a ruler, a quadratic function is like a curve, an exponential function is like a magnifying glass, and a logarithmic function is like a telescope.

Common Misconceptions:

❌ Students often confuse polynomial functions with rational functions.
✓ A polynomial function consists of terms with non-negative integer exponents, while a rational function is a ratio of two polynomials.
Why this confusion happens: Students may focus on the presence of variables in the denominator without recognizing the overall structure of the function.

Visual Description:

Linear: A straight line.
Quadratic: A parabola (U-shaped curve).
Polynomial: A smooth, continuous curve that can have multiple turning points. The end behavior (what happens as x approaches positive or negative infinity) is determined by the leading term (the term with the highest power of x).
Rational: Can have vertical asymptotes (where the denominator is zero), horizontal asymptotes (determined by the degrees of the numerator and denominator), and holes (where factors cancel out).
Exponential: Increases rapidly (if the base is greater than 1) or decreases rapidly (if the base is between 0 and 1). Has a horizontal asymptote at y = 0.
Logarithmic: Increases slowly. Has a vertical asymptote at x = 0.

Practice Check:

Question: Identify the type of function and describe its key characteristics: f(x) = (x² + 1)/(x - 2).

Answer: This is a rational function. It has a vertical asymptote at x = 2. Since the degree of the numerator (2) is one greater than the degree of the denominator (1), it has a slant asymptote.

Connection to Other Sections:

Identifying the type of function is crucial for applying the appropriate techniques for analyzing its behavior, transforming its graph, and solving related problems. For example, knowing that a function is quadratic allows you to use the quadratic formula to find its roots.

### 4.5 Transformations of Functions

Overview: Transformations allow us to manipulate the graph of a function by shifting, stretching, reflecting, or compressing it. Understanding transformations is crucial for visualizing function behavior and for creating new functions from existing ones.

The Core Concept: Transformations are operations that change the position, size, or orientation of a function's graph. The main types of transformations are:

Vertical Shifts: f(x) + c shifts the graph c units upward if c > 0, and c units downward if c < 0.
Horizontal Shifts: f(x - c) shifts the graph c units to the right if c > 0, and c units to the left if c < 0. Note the counter-intuitive sign.
Vertical Stretches and Compressions: c \ f(x) stretches the graph vertically by a factor of c if c > 1, and compresses it vertically by a factor of c if 0 < c < 1.
Horizontal Stretches and Compressions: f(c \ x) compresses the graph horizontally by a factor of c if c > 1, and stretches it horizontally by a factor of c if 0 < c < 1. Note the counter-intuitive effect.
Reflections about the x-axis: -f(x) reflects the graph about the x-axis.
Reflections about the y-axis: f(-x) reflects the graph about the y-axis.

Concrete Examples:

Example 1: Vertical and Horizontal Shifts
Original Function: f(x) = x²
Transformed Function: g(x) = (x - 2)² + 3
Analysis: The graph of g(x) is obtained by shifting the graph of f(x) 2 units to the right and 3 units upward.
Example 2: Vertical Stretch and Reflection
Original Function: f(x) = √x
Transformed Function: h(x) = -2√x
Analysis: The graph of h(x) is obtained by stretching the graph of f(x) vertically by a factor of 2 and reflecting it about the x-axis.
Example 3: Horizontal Compression
Original Function: f(x) = sin(x)
Transformed Function: k(x) = sin(2x)
Analysis: The graph of k(x) is obtained by compressing the graph of f(x) horizontally by a factor of 2. This means the period of the sine wave is halved.

Analogies & Mental Models:

Think of it like manipulating a rubber sheet: Shifting is like sliding the sheet, stretching is like pulling the sheet, reflecting is like flipping the sheet over.
Think of it like using Photoshop filters: Each transformation is like applying a different filter to an image.

Common Misconceptions:

❌ Students often confuse horizontal shifts with horizontal stretches/compressions and get the direction wrong.
✓ Remember that f(x - c) shifts the graph to the right if c > 0, and f(c \ x) compresses the graph horizontally by a factor of c if c > 1.
Why this confusion happens: The horizontal transformations act in the opposite direction to what students might intuitively expect.

Visual Description:

Visualize how each transformation affects the key features of the graph:
Vertical shifts: Move the entire graph up or down.
Horizontal shifts: Move the entire graph left or right.
Vertical stretches/compressions: Change the height of the graph.
Horizontal stretches/compressions: Change the width of the graph.
Reflections about the x-axis: Flip the graph over the x-axis.
Reflections about the y-axis: Flip the graph over the y-axis.

Practice Check:

Question: Describe the transformations applied to the function f(x) = x³ to obtain the function g(x) = - (x + 1)³ - 2.

Answer: The graph of g(x) is obtained by shifting the graph of f(x) 1 unit to the left, reflecting it about the x-axis, and shifting it 2 units downward.

Connection to Other Sections:

Understanding transformations is crucial for graphing functions, analyzing their behavior, and solving real-world problems. It also provides a deeper understanding of how different functions are related to each other.

### 4.6 Operations on Functions

Overview: Just like numbers, functions can be combined using arithmetic operations such as addition, subtraction, multiplication, and division. In addition, functions can be combined through a process called composition. Understanding these operations is crucial for building more complex functions from simpler ones and for modeling real-world phenomena.

The Core Concept:

Addition: (f + g)(x) = f(x) + g(x). The domain of f + g is the intersection of the domains of f and g.
Subtraction: (f - g)(x) = f(x) - g(x). The domain of f - g is the intersection of the domains of f and g.
Multiplication: (f \ g)(x) = f(x) \ g(x). The domain of f \ g is the intersection of the domains of f and g.
Division: (f / g)(x) = f(x) / g(x). The domain of f / g is the intersection of the domains of f and g, excluding any values of x for which g(x) = 0.
Composition: (f ∘ g)(x) = f(g(x)). This means you first apply the function g to x, and then apply the function f to the result. The domain of f ∘ g is the set of all x in the domain of g such that g(x) is in the domain of f.

Concrete Examples:

Example 1: Addition and Subtraction
f(x) = x² + 1 and g(x) = 2x - 3
(f + g)(x) = (x² + 1) + (2x - 3) = x² + 2x - 2
(f - g)(x) = (x² + 1) - (2x - 3) = x² - 2x + 4
Example 2: Multiplication and Division

Okay, here is a comprehensive pre-calculus lesson on functions, designed to be thorough, engaging, and suitable for high school students (grades 9-12). I've structured it with a focus on clarity, real-world applications, and building a strong foundation for future mathematical studies.

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## 1. INTRODUCTION

### 1.1 Hook & Context

Imagine you're designing a rollercoaster. The height of the track at any point depends on how far along the track the car has traveled. Or think about a self-driving car: the amount of steering required depends on its current position and speed. These scenarios, seemingly disparate, share a common mathematical thread: functions. Functions are fundamental to describing relationships where one thing depends on another. Have you ever used a vending machine? You press a button (the input) and receive a snack (the output). That's a function in action!

We use functions every day, often without realizing it. Consider the price of gas – it's a function of supply and demand. The temperature outside is a function of the time of day (and many other factors!). Even the grades you receive are (hopefully!) a function of the effort you put into your studies. Understanding functions gives us a powerful tool for modeling and predicting behavior in the world around us.

### 1.2 Why This Matters

Functions are the bedrock of advanced mathematics. They're essential for calculus, differential equations, linear algebra, and beyond. In the real world, functions are used in fields like engineering (designing bridges and circuits), economics (modeling market trends), computer science (writing algorithms), physics (describing motion), and medicine (analyzing drug dosages). A solid understanding of functions unlocks the ability to analyze complex systems and solve real-world problems.

This knowledge builds on prior algebra skills, particularly understanding variables, expressions, and equations. This lesson provides a strong foundation for exploring more advanced function types, such as trigonometric, exponential, and logarithmic functions. This will also prepare you for studying rates of change, optimization problems, and other key concepts in calculus.

### 1.3 Learning Journey Preview

In this lesson, we'll journey through the world of functions. We'll start by defining what a function is and how it differs from a relation. We'll learn how to represent functions using equations, graphs, tables, and verbal descriptions. We'll explore function notation and how to evaluate functions. We'll then delve into the domain and range of a function, and how to determine them. Next, we will cover function transformations and how they affect the graph of a function. We'll also investigate different types of functions, such as linear, quadratic, polynomial, and rational functions. Finally, we'll look at operations you can perform on functions, such as addition, subtraction, multiplication, division, and composition. Each concept will build on the previous one, equipping you with a comprehensive understanding of functions and their applications.

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## 2. LEARNING OBJECTIVES

By the end of this lesson, you will be able to:

1. Define a function and differentiate it from a relation using various representations (equations, graphs, tables, mappings).
2. Use function notation to represent and evaluate functions for given inputs.
3. Determine the domain and range of functions represented algebraically and graphically.
4. Identify and analyze different types of functions, including linear, quadratic, polynomial, and rational functions.
5. Graph functions using transformations (translations, reflections, stretches, and compressions).
6. Perform operations on functions, including addition, subtraction, multiplication, division, and composition.
7. Model real-world scenarios using functions and interpret the results in context.
8. Analyze the characteristics of functions, including intercepts, intervals of increasing/decreasing behavior, and end behavior.

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## 3. PREREQUISITE KNOWLEDGE

Before diving into functions, you should be comfortable with the following concepts:

Variables and Expressions: Understanding how to use variables to represent unknown quantities and manipulate algebraic expressions.
Equations: Solving linear and quadratic equations.
Coordinate Plane: Plotting points and understanding the relationship between points and their coordinates.
Graphing Lines: Knowing how to graph linear equations in slope-intercept form (y = mx + b).
Interval Notation: Representing sets of numbers using interval notation (e.g., [a, b], (a, b), (-∞, ∞)).
Basic Set Theory: Understanding the concept of a set and set notation.

If you need a refresher on any of these topics, I suggest reviewing your algebra textbook or using online resources like Khan Academy or Paul's Online Math Notes. These resources can provide a solid foundation for understanding functions.

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## 4. MAIN CONTENT

### 4.1 What is a Function?

Overview: A function is a special type of relationship between two sets, where each input from the first set (called the domain) is related to exactly one output in the second set (called the range). This "one-to-one" or "many-to-one" mapping is the core idea.

The Core Concept: At its heart, a function is a rule that assigns a unique output to each input. Imagine a machine: you put something in (the input), and the machine processes it according to its rule and spits out something else (the output). A function is similar. The key difference between a function and a general relation is this uniqueness requirement. In a relation, one input can potentially lead to multiple outputs. In a function, that's not allowed.

Think of it this way: If you input the same value into the function every time, you should get the same output every time. If that's not the case, it's not a function. A function is a well-defined relationship, meaning its behavior is predictable and consistent. We often represent functions using equations (like y = x^2), graphs (visual representations of the input-output pairs), tables (listing specific input-output values), or verbal descriptions (describing the rule in words).

The set of all possible inputs to a function is called the domain, and the set of all possible outputs is called the range. The domain is the set of values for which the function is defined, while the range is the set of values that the function can actually produce. Understanding the domain and range is crucial for understanding the behavior of a function.

Concrete Examples:

Example 1: Vending Machine
Setup: A vending machine has buttons labeled A1, A2, B1, B2, etc. Each button corresponds to a specific snack.
Process: You press the button "A1" (the input). The machine dispenses the snack associated with that button (the output).
Result: If "A1" always dispenses a bag of chips, then this is a function. If "A1" sometimes dispenses chips, sometimes candy, and sometimes nothing at all, then it's not a function.
Why this matters: This illustrates the "unique output" requirement. Each input (button) must lead to a single, predictable output (snack).

Example 2: Squaring Function (y = x^2)
Setup: Consider the equation y = x^2. We can input any real number for 'x'.
Process: If we input x = 2, we calculate y = 2^2 = 4. If we input x = -2, we calculate y = (-2)^2 = 4.
Result: Each input 'x' produces a single, unique output 'y'. Therefore, y = x^2 is a function.
Why this matters: This shows how an equation can represent a function. It also highlights that different inputs can lead to the same output (many-to-one), but each input must have only one output.

Analogies & Mental Models:

Think of it like a sorting machine: You feed different shapes into a machine, and it sorts them into different bins based on their properties (size, color, etc.). Each shape goes into only one bin. The shapes are the inputs, and the bins are the outputs.
Think of it like a recipe: You provide the ingredients (inputs), and the recipe (function) tells you how to combine them to create a specific dish (output). Using the same ingredients will always give you the same dish (if you follow the recipe correctly).

Common Misconceptions:

❌ Students often think that a function must be represented by an equation.
✓ Actually, a function can be represented by an equation, a graph, a table, or even a verbal description. The key is that each input has only one output.
Why this confusion happens: Equations are a common way to represent functions, but they are not the only way.

Visual Description:

Imagine a graph on the coordinate plane. For a graph to represent a function, it must pass the vertical line test. This means that if you draw any vertical line on the graph, the line should intersect the graph at most once. If a vertical line intersects the graph more than once, it means that one input (x-value) is associated with multiple outputs (y-values), and therefore it's not a function.

Practice Check:

Is the relation defined by the set of ordered pairs {(1, 2), (2, 4), (3, 6), (1, 3)} a function? Why or why not?

Answer: No, it is not a function. The input 1 is associated with two different outputs, 2 and 3. This violates the definition of a function.

Connection to Other Sections:

This section lays the foundation for understanding all subsequent sections. The definition of a function is crucial for understanding function notation (4.2), domain and range (4.3), and different types of functions (4.4).

### 4.2 Function Notation

Overview: Function notation is a standard way of representing functions using symbols. It provides a concise and efficient way to express the relationship between inputs and outputs.

The Core Concept: Instead of writing "y is a function of x," we use the notation f(x), which is read as "f of x." The letter 'f' represents the name of the function, and 'x' represents the input variable. f(x) represents the output of the function when the input is 'x'. For example, if f(x) = x^2 + 1, then f(3) means we substitute x = 3 into the equation, so f(3) = 3^2 + 1 = 10. The output is 10.

Function notation allows us to clearly identify the input and output values of a function. It also makes it easier to perform operations on functions and to compose functions (which we'll discuss later). We can use different letters to represent different functions, such as g(x), h(x), etc. This helps us distinguish between multiple functions in the same problem.

Concrete Examples:

Example 1: Evaluating f(x) = 2x + 3
Setup: We have the function f(x) = 2x + 3.
Process: To find f(4), we substitute x = 4 into the equation: f(4) = 2(4) + 3 = 8 + 3 = 11.
Result: f(4) = 11. This means that when the input is 4, the output of the function is 11.
Why this matters: This demonstrates how to use function notation to evaluate a function for a specific input.

Example 2: Evaluating g(t) = t^2 - 5
Setup: We have the function g(t) = t^2 - 5.
Process: To find g(-2), we substitute t = -2 into the equation: g(-2) = (-2)^2 - 5 = 4 - 5 = -1.
Result: g(-2) = -1. This means that when the input is -2, the output of the function is -1.
Why this matters: This shows that we can use different variables (like 't' instead of 'x') in function notation.

Analogies & Mental Models:

Think of f(x) as a "function machine": You put 'x' into the machine, and it performs some operations on 'x' to produce f(x).
Think of f(a) as "the value of the function f when x = a": It's simply a shorthand way of saying that.

Common Misconceptions:

❌ Students often think that f(x) means "f times x".
✓ Actually, f(x) means "the value of the function f at x". It's a single entity representing the output of the function.
Why this confusion happens: The parentheses can be misleading, as they are also used for multiplication.

Visual Description:

Imagine a graph of a function y = f(x). For any x-value on the x-axis, you can find the corresponding y-value (which is f(x)) on the y-axis. The function notation tells you that the y-coordinate of the point on the graph is equal to f(x).

Practice Check:

If h(x) = 3x - 1, find h(0) and h(a + 1).

Answer: h(0) = 3(0) - 1 = -1. h(a + 1) = 3(a + 1) - 1 = 3a + 3 - 1 = 3a + 2.

Connection to Other Sections:

Function notation is used extensively in all subsequent sections. It's essential for understanding how to evaluate functions, determine their domain and range, and perform operations on them.

### 4.3 Domain and Range

Overview: The domain and range are fundamental properties of a function that define its scope and possible outputs.

The Core Concept: The domain of a function is the set of all possible input values (x-values) for which the function is defined. In other words, it's the set of all values you can "plug in" to the function without causing any mathematical errors (like dividing by zero or taking the square root of a negative number).

The range of a function is the set of all possible output values (y-values) that the function can produce. It's the set of all values that the function can "spit out" after you've plugged in all the possible input values from the domain.

Determining the domain and range is crucial for understanding the behavior of a function. It tells you what values the function can accept and what values it can produce. The domain and range can be expressed using interval notation, set notation, or graphically.

Concrete Examples:

Example 1: f(x) = √(x - 2)
Setup: We have the function f(x) = √(x - 2).
Process: The square root function is only defined for non-negative numbers. Therefore, we must have x - 2 ≥ 0, which means x ≥ 2. The smallest possible output is 0.
Result: The domain is [2, ∞) and the range is [0, ∞).
Why this matters: This demonstrates how to find the domain and range of a function involving a square root.

Example 2: g(x) = 1/(x + 3)
Setup: We have the function g(x) = 1/(x + 3).
Process: Division by zero is undefined. Therefore, we must have x + 3 ≠ 0, which means x ≠ -3. The function can take on any value except 0.
Result: The domain is (-∞, -3) ∪ (-3, ∞) and the range is (-∞, 0) ∪ (0, ∞).
Why this matters: This shows how to find the domain and range of a function involving a fraction.

Analogies & Mental Models:

Think of the domain as the "allowed ingredients" for a recipe: You can only use ingredients that are listed in the recipe.
Think of the range as the "possible dishes" you can make with the recipe: The recipe can only produce certain dishes.

Common Misconceptions:

❌ Students often forget to consider restrictions on the domain, such as division by zero or square roots of negative numbers.
✓ Actually, you must always check for these restrictions when determining the domain of a function.
Why this confusion happens: It requires careful analysis of the function's equation.

Visual Description:

Imagine the graph of a function. The domain is the projection of the graph onto the x-axis (the set of all x-values covered by the graph). The range is the projection of the graph onto the y-axis (the set of all y-values covered by the graph).

Practice Check:

Find the domain and range of the function h(x) = x^2 + 1.

Answer: The domain is (-∞, ∞) since you can square any real number. The range is [1, ∞) because the smallest value of x^2 is 0, so the smallest value of x^2 + 1 is 1.

Connection to Other Sections:

Understanding the domain and range is essential for graphing functions (4.5) and analyzing their behavior (4.8).

### 4.4 Types of Functions

Overview: Functions come in various forms, each with unique characteristics and behaviors. Understanding these types is crucial for modeling real-world situations.

The Core Concept: We can classify functions into different categories based on their equations and graphs. Some common types include:

Linear Functions: Functions of the form f(x) = mx + b, where m is the slope and b is the y-intercept. Their graphs are straight lines.
Quadratic Functions: Functions of the form f(x) = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0. Their graphs are parabolas.
Polynomial Functions: Functions of the form f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where a_n, a_{n-1}, ..., a_1, a_0 are constants and n is a non-negative integer. Their graphs are smooth curves.
Rational Functions: Functions of the form f(x) = p(x)/q(x), where p(x) and q(x) are polynomials and q(x) ≠ 0. Their graphs can have asymptotes (lines that the graph approaches but never touches).
Exponential Functions: Functions of the form f(x) = a^x, where a is a positive constant and a ≠ 1. Their graphs exhibit exponential growth or decay.
Logarithmic Functions: Functions of the form f(x) = log_a(x), where a is a positive constant and a ≠ 1. They are the inverse of exponential functions.

Understanding the characteristics of each type of function allows us to predict its behavior and use it to model real-world phenomena.

Concrete Examples:

Example 1: Linear Function (f(x) = 2x + 1)
Setup: f(x) = 2x + 1 is a linear function with a slope of 2 and a y-intercept of 1.
Process: We can plot two points (e.g., (0, 1) and (1, 3)) and draw a straight line through them.
Result: The graph is a straight line that rises from left to right.
Why this matters: Linear functions are used to model constant rates of change.

Example 2: Quadratic Function (g(x) = x^2 - 4x + 3)
Setup: g(x) = x^2 - 4x + 3 is a quadratic function.
Process: We can find the vertex of the parabola (x = -b/2a = 2), and the x-intercepts (x = 1 and x = 3).
Result: The graph is a parabola that opens upwards, with a vertex at (2, -1) and x-intercepts at (1, 0) and (3, 0).
Why this matters: Quadratic functions are used to model projectile motion and optimization problems.

Analogies & Mental Models:

Linear Function: Think of a constant speed. For every unit of time, you cover the same distance.
Quadratic Function: Think of a ball thrown into the air. Its height increases and then decreases, forming a parabolic path.

Common Misconceptions:

❌ Students often confuse polynomial functions with rational functions.
✓ Actually, polynomial functions are sums of terms with non-negative integer exponents, while rational functions are ratios of polynomials.
Why this confusion happens: Both involve polynomials, but the structure is different.

Visual Description:

Each type of function has a distinct graph. Linear functions have straight lines, quadratic functions have parabolas, and so on. Understanding the shape of the graph is crucial for recognizing the type of function.

Practice Check:

Identify the type of function: h(x) = (x^3 + 1)/(x - 2).

Answer: This is a rational function because it is a ratio of two polynomials.

Connection to Other Sections:

Understanding different types of functions is essential for graphing them (4.5), analyzing their behavior (4.8), and modeling real-world situations (7).

### 4.5 Graphing Functions Using Transformations

Overview: Transformations allow us to manipulate the graph of a function by shifting, stretching, compressing, or reflecting it.

The Core Concept: Transformations are changes that we can apply to a function's graph to create a new graph. The most common types of transformations are:

Vertical Translations: Shifting the graph up or down. f(x) + c shifts the graph up by c units; f(x) - c shifts the graph down by c units.
Horizontal Translations: Shifting the graph left or right. f(x - c) shifts the graph right by c units; f(x + c) shifts the graph left by c units.
Vertical Stretches and Compressions: Stretching or compressing the graph vertically. c f(x) stretches the graph vertically by a factor of c if c > 1, and compresses it vertically by a factor of c if 0 < c < 1.
Horizontal Stretches and Compressions: Stretching or compressing the graph horizontally. f(c x) compresses the graph horizontally by a factor of c if c > 1, and stretches it horizontally by a factor of c if 0 < c < 1.
Reflections: Reflecting the graph across the x-axis or y-axis. -f(x) reflects the graph across the x-axis; f(-x) reflects the graph across the y-axis.

Understanding how these transformations affect the graph of a function allows us to sketch the graph quickly and easily.

Concrete Examples:

Example 1: Transforming f(x) = x^2
Setup: We start with the graph of f(x) = x^2 (a parabola).
Process: To graph g(x) = (x - 2)^2 + 3, we first shift the graph of f(x) 2 units to the right (horizontal translation) and then shift it 3 units up (vertical translation).
Result: The graph of g(x) is a parabola with its vertex at (2, 3).
Why this matters: This demonstrates how to combine multiple transformations to create a new graph.

Example 2: Transforming f(x) = |x|
Setup: We start with the graph of f(x) = |x| (a V-shaped graph).
Process: To graph h(x) = -2|x + 1|, we first shift the graph of f(x) 1 unit to the left (horizontal translation), then stretch it vertically by a factor of 2, and finally reflect it across the x-axis.
Result: The graph of h(x) is an inverted V-shaped graph with its vertex at (-1, 0).
Why this matters: This shows how to use reflections and stretches/compressions in addition to translations.

Analogies & Mental Models:

Think of transformations as "editing" the graph of a function: You can move it around, stretch it, compress it, or flip it over.
Think of translations as "sliding" the graph: You're moving it without changing its shape.
Think of stretches/compressions as "resizing" the graph: You're making it taller/shorter or wider/narrower.
Think of reflections as "mirroring" the graph: You're flipping it over a line.

Common Misconceptions:

❌ Students often get horizontal translations backwards: f(x - c) shifts the graph to the right, not the left.
✓ Actually, f(x - c) shifts the graph to the right because you need to increase x by c to get the same output as f(x).
Why this confusion happens: The minus sign can be counterintuitive.

Visual Description:

Visualize the graph of the original function and then imagine how each transformation changes its shape and position.

Practice Check:

Describe the transformations needed to graph g(x) = -(x + 3)^2 - 1 from the graph of f(x) = x^2.

Answer: Shift 3 units to the left, reflect across the x-axis, and shift 1 unit down.

Connection to Other Sections:

Understanding transformations is essential for graphing different types of functions (4.4) and analyzing their behavior (4.8).

### 4.6 Operations on Functions

Overview: Just like we can perform arithmetic operations on numbers, we can also perform operations on functions, creating new functions.

The Core Concept: We can add, subtract, multiply, and divide functions, as well as compose them.

Addition: (f + g)(x) = f(x) + g(x)
Subtraction: (f - g)(x) = f(x) - g(x)
Multiplication: (f g)(x) = f(x) g(x)
Division: (f / g)(x) = f(x) / g(x), where g(x) ≠ 0
Composition: (f ∘ g)(x) = f(g(x)). This means we first apply the function g to x, and then apply the function f to the result.

The domain of the resulting function depends on the domains of the original functions and any restrictions imposed by the operation (e.g., division by zero). Composition requires special attention, as the output of the inner function (g(x)) must be in the domain of the outer function (f(x)).

Concrete Examples:

Example 1: f(x) = x + 1, g(x) = x^2
Setup: We have two functions, f(x) = x + 1 and g(x) = x^2.
Process:
(f + g)(x) = (x + 1) + x^2 = x^2 + x + 1
(f - g)(x) = (x + 1) - x^2 = -x^2 + x + 1
(f g)(x) = (x + 1) x^2 = x^3 + x^2
(f / g)(x) = (x + 1) / x^2, where x ≠ 0
(f ∘ g)(x) = f(g(x)) = f(x^2) = x^2 + 1
(g ∘ f)(x) = g(f(x)) = g(x + 1) = (x + 1)^2 = x^2 + 2x + 1
Result: We have found the sum, difference, product, quotient, and composition of the two functions.
Why this matters: This demonstrates how to perform different operations on functions.

Example 2: Domain of (f / g)(x)
Setup: Let f(x) = √(x + 2) and g(x) = x - 1.
Process: (f/g)(x) = √(x+2)/(x-1). The domain of f(x) is x ≥ -2. The domain of g(x) is all real numbers. However, we must also ensure x-1 ≠ 0, so x ≠ 1.
Result: The domain of (f/g)(x) is [-2, 1) ∪ (1, ∞).
Why this matters: This highlights the importance of considering the domains of both functions when performing operations, especially division.

Analogies & Mental Models:

Think of function operations as "combining" different recipes: You can add ingredients from different recipes, subtract ingredients, multiply them, or even combine them in a more complex way (like composition).
Think of composition as "chaining" functions together: The output of one function becomes the input of another.

Common Misconceptions:

❌ Students often confuse (f ∘ g)(x) with (f g)(x).
✓ Actually, (f ∘ g)(x) = f(g(x)) is composition, while (f g)(x) = f(x) g(x) is multiplication.
Why this confusion happens: The notation is similar, but the operations are very different.

Visual Description:

There isn't a simple visual representation for all function operations. Composition can be visualized as a flow chart: x goes into g, g(x) comes out, and then g(x) goes into f, and f(g(x)) comes out.

Practice Check:

If f(x) = 2x and g(x) = x - 3, find (f ∘ g)(x) and (g ∘ f)(x).

Answer: (f ∘ g)(x) = f(g(x)) = f(x - 3) = 2(x - 3) = 2x - 6. (g ∘ f)(x) = g(f(x)) = g(2x) = 2x - 3.

Connection to Other Sections:

Understanding operations on functions is essential for building more complex functions and modeling real-world situations.

### 4.7 Modeling with Functions

Overview: Functions provide a powerful tool for representing and analyzing real-world phenomena.

The Core Concept: We can use functions to model relationships between variables in various real-world scenarios. This involves identifying the independent and dependent variables, choosing an appropriate type of function, and determining the parameters of the function.

For example, we can use a linear function to model the cost of renting a car based on the number of miles driven, a quadratic function to model the trajectory of a projectile, or an exponential function to model population growth.

Once we have a function that models a real-world situation, we can use it to make predictions, solve problems, and gain insights into the behavior of the system.

Concrete Examples:

Example 1: Linear Model (Cost of a Taxi Ride)
Setup: A taxi charges a flat fee of $3 plus $2 per mile.
Process: Let x be the number of miles driven and y be the total cost. The function that models this situation is y = 2x + 3.
Result: We can use this function to find the cost of a 5-mile ride: y = 2(5) + 3 = $13.
Why this matters: This demonstrates how to use a linear function to model a simple real-world scenario.

Example 2: Quadratic Model (Projectile Motion)
Setup: A ball is thrown upwards with an initial velocity of 48 feet per second from a height of 6 feet. The height of the ball after t seconds is given by h(t) = -16t^2 + 48t + 6.
Process: We can use this function to find the maximum height of the ball and the time it takes to reach the ground.
Result: The maximum height is reached at t = 1.5 seconds, and the maximum height is h(1.5) = 42 feet.
Why this matters: This shows how to use a quadratic function to model projectile motion.

Analogies & Mental Models:

Think of modeling as "creating a mathematical representation" of a real-world situation: You're using functions to capture the relationships between variables.
Think of a function as a "translator" between the real world and the mathematical world: It allows you to translate real-world problems into mathematical equations and then translate the solutions back into real-world answers.

Common Misconceptions:

❌ Students often struggle to choose the appropriate type of function for a given real-world situation.
✓ Actually, you need to carefully analyze the relationships between the variables and consider the characteristics of different types of functions.
Why this confusion happens: It requires experience and a good understanding of different types of functions.

Visual Description:

Graphs of functions can be used to visualize real-world situations. For example, the graph of a linear function can represent the cost of a product over time, and the graph of a quadratic function can represent the trajectory of a projectile.

Practice Check:

A store sells t-shirts for $10 each. Write a function that models the revenue generated from selling x t-shirts.

Answer: R(x) = 10x, where R(x) is the revenue and x is the number of t-shirts sold.

Connection to Other Sections:

Modeling with functions requires a solid understanding of different types of functions (4.4), graphing (4.5), and operations on functions (4.6).

### 4.8 Analyzing Function Characteristics

Overview: Analyzing the characteristics of a function provides a deeper understanding of its behavior and properties.

The Core Concept: Key characteristics of functions include:

Intercepts: The points where the graph of the function intersects the x-axis (x-intercepts) and the y-axis (y-intercept).
Intervals of Increasing/Decreasing Behavior: The intervals over which the function is increasing (y-values are increasing as x-values increase) or decreasing (y-values are decreasing as x-values increase).
Maximum and Minimum Values: The highest and lowest points on

Okay, here's a comprehensive pre-calculus lesson on Functions. I've aimed for depth, clarity, and engagement, structuring it to build understanding progressively.

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## 1. INTRODUCTION

### 1.1 Hook & Context

Imagine you're building a rollercoaster. You need to know exactly how high each point should be, how steep the drops can be, and how fast the cars will travel at any given moment. Or think about designing a bridge – the engineers have to be able to predict how the bridge will respond to different weights and stresses. These scenarios, seemingly complex, rely heavily on a fundamental mathematical concept: functions. Functions are the workhorses that model relationships between different quantities, allowing us to predict outcomes and design solutions. Think of a simple vending machine - you put in a specific amount of money (the input), and you get a specific snack (the output). That's a function in action!

Think about your phone. When you tap an app icon, the phone performs a series of operations to launch the app. The tap is the input, and the app launching is the output. Or consider online shopping – you search for a product (input), and the website returns a list of relevant items (output). Functions are all around us, even if we don't always realize it. This lesson will peel back the layers and reveal the power and versatility of functions.

### 1.2 Why This Matters

Understanding functions is crucial for success in pre-calculus and calculus, which in turn are essential for many STEM fields. Whether you're interested in engineering, computer science, physics, economics, or even data science, a solid grasp of functions will be your foundation. Functions allow us to model real-world phenomena, analyze data, and make predictions. This knowledge builds upon your prior understanding of algebra, particularly equations and graphing, and sets the stage for more advanced topics like limits, derivatives, and integrals. In essence, mastering functions unlocks a deeper understanding of how the world works and provides the tools to solve complex problems.

Moreover, understanding functions has direct career applications. Engineers use functions to model the behavior of circuits and structures. Economists use functions to analyze market trends and predict economic growth. Computer scientists use functions to write efficient and reliable code. Data scientists use functions to create models that can predict customer behavior or identify fraud. The possibilities are virtually endless.

### 1.3 Learning Journey Preview

In this lesson, we will begin by defining what a function is and how it differs from a relation. We will then explore different ways to represent functions: graphically, algebraically, numerically (tables), and verbally. We will delve into function notation and learn how to evaluate functions for specific inputs. We’ll then examine key characteristics of functions, such as domain, range, intercepts, symmetry, and increasing/decreasing intervals. After mastering these fundamental concepts, we will learn about different types of functions, including linear, quadratic, polynomial, rational, exponential, and logarithmic functions. Finally, we will explore transformations of functions and how to combine functions through arithmetic operations and composition. By the end of this journey, you will have a solid understanding of functions and be able to apply this knowledge to solve a wide range of problems.

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## 2. LEARNING OBJECTIVES

By the end of this lesson, you will be able to:

1. Define a function and differentiate it from a relation using the vertical line test and mapping diagrams.
2. Represent functions in multiple ways (algebraically, graphically, numerically, and verbally) and translate between these representations.
3. Evaluate functions using function notation, including finding f(a), f(x+h), and piecewise functions.
4. Determine the domain and range of a function from its equation, graph, or table.
5. Identify key characteristics of a function, including intercepts, symmetry (even/odd), intervals of increasing and decreasing behavior, and extrema (maxima/minima).
6. Analyze and graph linear, quadratic, polynomial, rational, exponential, and logarithmic functions, identifying their key features.
7. Apply transformations (translations, reflections, stretches, and compressions) to the graph of a function and write the equation of the transformed function.
8. Combine functions using arithmetic operations (addition, subtraction, multiplication, division) and function composition, and determine the domain of the resulting function.

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## 3. PREREQUISITE KNOWLEDGE

Before diving into functions, you should have a solid understanding of the following concepts:

Basic Algebra: Solving equations, simplifying expressions, working with exponents and radicals.
Coordinate Plane: Plotting points, understanding x- and y-axes, quadrants.
Linear Equations: Slope-intercept form (y = mx + b), point-slope form, finding the equation of a line given two points or a point and a slope.
Graphing Basics: Plotting points and connecting them to create a graph.
Sets and Set Notation: Understanding sets, subsets, unions, intersections, and interval notation.
Inequalities: Solving and graphing inequalities on a number line.

If you need a refresher on any of these topics, consider reviewing your algebra textbook or online resources like Khan Academy or Paul's Online Math Notes. A strong foundation in these areas will make learning about functions much smoother.

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## 4. MAIN CONTENT

### 4.1 What is a Function?

Overview: A function is a special type of relation that maps each input to exactly one output. It's a fundamental concept in mathematics, describing how one quantity depends on another.

The Core Concept: In simple terms, a function is a rule that assigns to each element in one set (called the domain) exactly one element in another set (called the range). Think of it as a machine: you put something in (the input), and the machine does something to it and gives you something out (the output). The key is that for each input, you always get the same output.

More formally, a relation is simply a set of ordered pairs (x, y). A function is a relation where each x-value is paired with only one y-value. This "one-to-one" or "many-to-one" mapping from x to y is what distinguishes a function from a general relation. A relation can have one x-value mapped to multiple y-values, but a function cannot.

We often use the notation f(x) to represent a function. This reads as "f of x" and means "the value of the function f at x." Here, 'x' is the input, and 'f(x)' is the output. The set of all possible input values (x-values) is called the domain of the function, and the set of all possible output values (f(x)-values) is called the range of the function.

The concept of a function is critical because it allows us to model and analyze relationships between variables. By understanding how one variable depends on another, we can make predictions, solve problems, and gain insights into the world around us.

Concrete Examples:

Example 1: The Area of a Circle
Setup: The area of a circle depends on its radius. We can define a function A(r) that gives the area of a circle with radius r.
Process: The formula for the area of a circle is A = πr². Therefore, A(r) = πr². If we input a radius of 2, we get A(2) = π(2²) = 4π.
Result: A(2) = 4π. This means a circle with a radius of 2 has an area of 4π (approximately 12.57).
Why this matters: This demonstrates how a function can model a real-world relationship – the relationship between a circle's radius and its area. For any given radius, there's only one possible area.

Example 2: A Vending Machine
Setup: Imagine a vending machine. You select a specific item (e.g., "A3").
Process: The machine processes your selection and dispenses the corresponding item.
Result: You get the item associated with "A3." If "A3" is soda, you get soda.
Why this matters: The vending machine acts as a function. The input is the item code, and the output is the specific item. Each code corresponds to only one item.

Analogies & Mental Models:

Think of it like a factory: Raw materials (the input) go into a factory, and the factory processes them to produce a finished product (the output). Each set of raw materials produces a specific, predictable product.
How the analogy maps to the concept: The factory represents the function, the raw materials represent the input (x-value), and the finished product represents the output (f(x)-value).
Where the analogy breaks down (limitations): Factories can sometimes have errors or breakdowns, producing unexpected results. Functions, in their mathematical definition, are deterministic and always produce the same output for the same input.

Common Misconceptions:

❌ Students often think that f(x) means "f times x."
✓ Actually, f(x) means "the value of the function f at x." It's a notation that indicates the output of the function for a given input.
Why this confusion happens: The notation is similar to multiplication, but it represents a different concept. It's important to understand that 'f' is the name of the function, and 'x' is the input.

Visual Description:

Imagine a diagram with two ovals. The left oval represents the domain (x-values), and the right oval represents the range (y-values). Arrows connect elements in the domain to elements in the range. For a relation to be a function, each element in the domain must have exactly one arrow pointing to an element in the range. It's okay for multiple elements in the domain to point to the same element in the range, but no element in the domain can point to multiple elements in the range. This is the vertical line test in graphical form. If you can draw a vertical line that intersects the graph of a relation more than once, it is NOT a function.

Practice Check:

Is the relation represented by the set of ordered pairs {(1, 2), (2, 4), (3, 6), (1, 8)} a function? Why or why not?

Answer: No, it is not a function because the x-value 1 is paired with two different y-values (2 and 8).

Connection to Other Sections:

This section lays the foundation for understanding all subsequent sections. The definition of a function is essential for understanding function notation, domain and range, types of functions, and transformations.

### 4.2 Representing Functions

Overview: Functions can be represented in several ways: algebraically (using an equation), graphically (using a graph), numerically (using a table), and verbally (using a description). Understanding these different representations is crucial for analyzing and working with functions.

The Core Concept: The four main ways to represent functions provide different perspectives and insights.

Algebraic Representation: This is the most common way to represent a function, using an equation that relates the input (x) to the output (f(x)). For example, f(x) = 2x + 1, g(x) = x², h(x) = √x. The algebraic representation allows us to easily calculate the output for any given input.

Graphical Representation: This involves plotting the ordered pairs (x, f(x)) on a coordinate plane. The graph of a function provides a visual representation of the relationship between the input and output. We can easily identify key features of the function, such as intercepts, maximums, minimums, and increasing/decreasing intervals.

Numerical Representation: This involves creating a table of values that shows the input (x) and the corresponding output (f(x)). The table provides a discrete set of data points that represent the function. This is particularly useful when the function is defined by a set of data points rather than an equation.

Verbal Representation: This involves describing the function in words. For example, "The function squares the input and adds 3." While not as precise as the other representations, the verbal representation can provide a conceptual understanding of the function.

Being able to translate between these representations is a key skill. For example, you should be able to take an equation and create a table of values, or take a graph and write an equation that represents it.

Concrete Examples:

Example 1: The Function f(x) = x²
Algebraic: f(x) = x²
Graphical: A parabola opening upwards, with its vertex at the origin (0, 0).
Numerical:
| x | f(x) |
| --- | ---- |
| -2 | 4 |
| -1 | 1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 4 |
Verbal: "The function squares the input."

Example 2: The Cost of Shipping
Algebraic: Let C(w) be the cost of shipping a package weighing 'w' pounds. Suppose the cost is $5 plus $1 per pound. Then C(w) = 5 + w.
Graphical: A straight line with a y-intercept of 5 and a slope of 1.
Numerical:
| w (pounds) | C(w) (dollars) |
| ---------- | --------------- |
| 1 | 6 |
| 2 | 7 |
| 3 | 8 |
| 4 | 9 |
Verbal: "The cost of shipping is $5 plus $1 for each pound."

Analogies & Mental Models:

Think of it like describing a landscape: You can describe a mountain using words (verbal), draw a picture of it (graphical), write down the coordinates of various points on the mountain (numerical), or write a mathematical equation that describes its shape (algebraic). All these representations describe the same mountain but in different ways.

Common Misconceptions:

❌ Students often struggle to connect the different representations. They might see the equation f(x) = x² and not immediately recognize the shape of the parabola on a graph.
✓ Actually, each representation provides a different perspective on the same underlying function. Practice translating between the representations to strengthen your understanding.
Why this confusion happens: Each representation requires a different skill set. Algebra requires symbolic manipulation, graphing requires visualization, and numerical representation requires data analysis.

Visual Description:

Imagine four windows, each showing a different view of the same object. One window shows the object's mathematical formula, another shows its shape, another shows a table of its dimensions, and the last describes it in words. The object is the function, and each window shows a different representation of it.

Practice Check:

Given the function f(x) = 3x - 2, create a table of values for x = -1, 0, 1, and 2, and then sketch a graph of the function.

Answer:

Table:
| x | f(x) |
| --- | ---- |
| -1 | -5 |
| 0 | -2 |
| 1 | 1 |
| 2 | 4 |

Graph: A straight line passing through the points (-1, -5), (0, -2), (1, 1), and (2, 4).

Connection to Other Sections:

Understanding the different representations of functions is essential for evaluating functions, determining domain and range, and analyzing different types of functions.

### 4.3 Function Notation and Evaluation

Overview: Function notation is a concise way to represent and work with functions. Evaluating a function means finding the output value for a given input value.

The Core Concept: Function notation, such as f(x), is a powerful tool for representing and manipulating functions. It allows us to clearly identify the input and output of a function, and to perform operations on functions in a concise and unambiguous way. The notation f(x) represents the value of the function f at the input x. The variable 'x' is a placeholder, and we can substitute any value or expression for 'x'.

Evaluating a function means finding the value of f(x) for a specific value of x. To evaluate a function, we simply substitute the given value of x into the function's equation and simplify. For example, if f(x) = x² + 1, then f(3) = 3² + 1 = 10.

We can also evaluate functions for expressions, such as f(x + h). In this case, we substitute the entire expression (x + h) for x in the function's equation. For example, if f(x) = x², then f(x + h) = (x + h)² = x² + 2xh + h². This type of evaluation is particularly important in calculus when finding derivatives.

Piecewise functions are functions that are defined by different equations for different intervals of the domain. To evaluate a piecewise function, we must first determine which interval the input value belongs to, and then use the corresponding equation to find the output value.

Concrete Examples:

Example 1: Evaluating f(x) = 2x + 3
Find f(4): f(4) = 2(4) + 3 = 8 + 3 = 11
Find f(-1): f(-1) = 2(-1) + 3 = -2 + 3 = 1
Find f(a): f(a) = 2(a) + 3 = 2a + 3
Find f(x + h): f(x + h) = 2(x + h) + 3 = 2x + 2h + 3

Example 2: Evaluating a Piecewise Function
Consider the piecewise function:
f(x) = { x² if x < 0
{ 3x if x >= 0
Find f(-2): Since -2 < 0, we use the equation f(x) = x². So, f(-2) = (-2)² = 4.
Find f(5): Since 5 >= 0, we use the equation f(x) = 3x. So, f(5) = 3(5) = 15.

Analogies & Mental Models:

Think of function notation like a recipe: The function f is the name of the recipe, x is the ingredient, and f(x) is the finished dish. Evaluating the function is like following the recipe to create the dish.

Common Misconceptions:

❌ Students often incorrectly distribute when evaluating f(x + h). For example, if f(x) = x², they might write f(x + h) = x² + h², forgetting the middle term 2xh.
✓ Actually, f(x + h) = (x + h)² = (x + h)(x + h) = x² + 2xh + h². Remember to use the FOIL method or the binomial theorem when expanding expressions.
Why this confusion happens: Distributing correctly requires careful attention to the order of operations and the rules of algebra.

Visual Description:

Imagine a function machine. You put in a number (x), the machine processes it according to the function's rule (f), and then it spits out a new number (f(x)). The function notation simply describes this process in a mathematical way.

Practice Check:

Given the function g(x) = x³ - 2x + 1, find g(2) and g(-1).

Answer:

g(2) = (2)³ - 2(2) + 1 = 8 - 4 + 1 = 5
g(-1) = (-1)³ - 2(-1) + 1 = -1 + 2 + 1 = 2

Connection to Other Sections:

Function evaluation is a fundamental skill that is used throughout pre-calculus and calculus. It is essential for graphing functions, finding intercepts, and solving equations.

### 4.4 Domain and Range

Overview: The domain of a function is the set of all possible input values, and the range is the set of all possible output values. Determining the domain and range is crucial for understanding the behavior of a function.

The Core Concept: The domain of a function is the set of all possible input values (x-values) for which the function is defined. In other words, it's the set of all x-values that you can plug into the function without causing any mathematical errors, such as division by zero or taking the square root of a negative number.

The range of a function is the set of all possible output values (y-values or f(x)-values) that the function can produce. In other words, it's the set of all y-values that you can get out of the function by plugging in all possible x-values from the domain.

To determine the domain of a function, we need to identify any restrictions on the input values. Common restrictions include:

Division by zero: The denominator of a fraction cannot be zero.
Square root of a negative number: The radicand (the expression under the square root) must be non-negative.
Logarithm of a non-positive number: The argument of a logarithm must be positive.

To determine the range of a function, we can use several methods, including:

Graphing the function: The range is the set of all y-values that the graph covers.
Analyzing the equation: We can often determine the range by considering the possible values of the function's output.
Using transformations: We can use transformations to determine how the range is affected by changes to the function's equation.

Concrete Examples:

Example 1: Finding the Domain and Range of f(x) = √x
Domain: The expression under the square root must be non-negative, so x >= 0. Therefore, the domain is [0, ∞).
Range: The square root of a non-negative number is always non-negative, so f(x) >= 0. Therefore, the range is [0, ∞).

Example 2: Finding the Domain and Range of g(x) = 1/(x - 2)
Domain: The denominator cannot be zero, so x - 2 ≠ 0, which means x ≠ 2. Therefore, the domain is (-∞, 2) ∪ (2, ∞).
Range: The function can take on any value except 0. Therefore, the range is (-∞, 0) ∪ (0, ∞).

Analogies & Mental Models:

Think of the domain as the ingredients list for a recipe: You can only use ingredients that are on the list. The range is the set of all possible dishes you can make using those ingredients.

Common Misconceptions:

❌ Students often forget to consider restrictions on the domain, especially when dealing with fractions or square roots.
✓ Actually, it's crucial to carefully examine the function's equation and identify any potential restrictions on the input values.
Why this confusion happens: Identifying restrictions requires a thorough understanding of algebraic rules and properties.

Visual Description:

Imagine the graph of a function. The domain is the projection of the graph onto the x-axis, and the range is the projection of the graph onto the y-axis.

Practice Check:

Find the domain and range of the function h(x) = x² + 3.

Answer:

Domain: There are no restrictions on the input values, so the domain is (-∞, ∞).
Range: The function is a parabola opening upwards with a vertex at (0, 3). Therefore, the range is [3, ∞).

Connection to Other Sections:

Understanding domain and range is essential for analyzing the behavior of functions, graphing functions, and solving equations. It is also crucial for understanding the concept of inverse functions.

### 4.5 Key Characteristics of Functions

Overview: Understanding the key characteristics of a function, such as intercepts, symmetry, increasing/decreasing intervals, and extrema, helps us analyze and interpret its behavior.

The Core Concept: Several key features help us understand the behavior of a function and visualize its graph:

Intercepts:
x-intercepts: The points where the graph intersects the x-axis. These are found by setting f(x) = 0 and solving for x. They are also known as roots or zeros of the function.
y-intercept: The point where the graph intersects the y-axis. This is found by setting x = 0 and evaluating f(0).

Symmetry:
Even Function: A function is even if f(-x) = f(x) for all x in the domain. The graph of an even function is symmetric with respect to the y-axis. Example: f(x) = x².
Odd Function: A function is odd if f(-x) = -f(x) for all x in the domain. The graph of an odd function is symmetric with respect to the origin. Example: f(x) = x³.
Neither Even Nor Odd: A function that does not satisfy either of the above conditions.

Increasing and Decreasing Intervals:
Increasing Interval: An interval where the function's values increase as x increases. The graph slopes upwards from left to right.
Decreasing Interval: An interval where the function's values decrease as x increases. The graph slopes downwards from left to right.
Constant Interval: An interval where the function's values remain constant as x increases. The graph is a horizontal line.

Extrema (Maxima and Minima):
Local Maximum: A point where the function's value is greater than or equal to the values at nearby points.
Local Minimum: A point where the function's value is less than or equal to the values at nearby points.
Absolute Maximum: The highest value of the function over its entire domain.
Absolute Minimum: The lowest value of the function over its entire domain.

Concrete Examples:

Example 1: Analyzing f(x) = x² - 4
x-intercepts: Set f(x) = 0: x² - 4 = 0 => x = ±2. The x-intercepts are (2, 0) and (-2, 0).
y-intercept: Set x = 0: f(0) = 0² - 4 = -4. The y-intercept is (0, -4).
Symmetry: f(-x) = (-x)² - 4 = x² - 4 = f(x). The function is even.
Increasing/Decreasing: The function is decreasing on the interval (-∞, 0) and increasing on the interval (0, ∞).
Extrema: The function has a local (and absolute) minimum at (0, -4).

Example 2: Analyzing g(x) = x³
x-intercept: Set g(x) = 0: x³ = 0 => x = 0. The x-intercept is (0, 0).
y-intercept: Set x = 0: g(0) = 0³ = 0. The y-intercept is (0, 0).
Symmetry: g(-x) = (-x)³ = -x³ = -g(x). The function is odd.
Increasing/Decreasing: The function is increasing on the interval (-∞, ∞).
Extrema: The function has no local or absolute maxima or minima.

Analogies & Mental Models:

Think of analyzing a function like reading a map: The intercepts are like landmarks, symmetry is like recognizing a pattern, increasing/decreasing intervals are like hills and valleys, and extrema are like the highest and lowest points on the map.

Common Misconceptions:

❌ Students often confuse even and odd functions. They might think that if a function isn't even, it must be odd.
✓ Actually, a function can be neither even nor odd. It's important to test both conditions to determine the function's symmetry.
Why this confusion happens: The definitions of even and odd functions can be tricky to remember, and it's easy to make mistakes when substituting -x into the function's equation.

Visual Description:

Imagine the graph of a function. The intercepts are the points where the graph crosses the axes. Symmetry is whether the graph is a mirror image across the y-axis (even) or the origin (odd). Increasing/decreasing intervals are the sections where the graph goes up or down. Extrema are the peaks and valleys of the graph.

Practice Check:

Determine the intercepts, symmetry, and intervals of increasing/decreasing behavior for the function h(x) = -x + 2.

Answer:

x-intercept: (2, 0)
y-intercept: (0, 2)
Symmetry: Neither even nor odd
Increasing/Decreasing: Decreasing on the interval (-∞, ∞)

Connection to Other Sections:

Understanding the key characteristics of functions is essential for graphing functions, solving equations, and modeling real-world phenomena.

### 4.6 Types of Functions

Overview: Pre-calculus introduces several important types of functions, each with its own unique properties and characteristics.

The Core Concept: Understanding different types of functions is essential for modeling real-world phenomena and solving mathematical problems. Here's an overview of some key function types:

Linear Functions: Functions of the form f(x) = mx + b, where m is the slope and b is the y-intercept. Their graphs are straight lines.
Quadratic Functions: Functions of the form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. Their graphs are parabolas.
Polynomial Functions: Functions of the form f(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀, where a_n, a_n-1, ..., a₁, a₀ are constants and n is a non-negative integer. Their graphs are smooth, continuous curves.
Rational Functions: Functions of the form f(x) = p(x) / q(x), where p(x) and q(x) are polynomial functions and q(x) ≠ 0. Their graphs can have vertical and horizontal asymptotes.
Exponential Functions: Functions of the form f(x) = a^x, where a is a positive constant and a ≠ 1. Their graphs exhibit exponential growth or decay.
Logarithmic Functions: Functions of the form f(x) = log_a(x), where a is a positive constant and a ≠ 1. They are the inverse functions of exponential functions.

Each type of function has specific characteristics and properties that make it suitable for modeling different types of phenomena. For example, linear functions are used to model constant rates of change, quadratic functions are used to model projectile motion, and exponential functions are used to model population growth.

Concrete Examples:

Example 1: Linear Function f(x) = 2x + 1
Graph: A straight line with a slope of 2 and a y-intercept of 1.
Properties: Constant rate of change, no asymptotes, domain and range are both (-∞, ∞).

Example 2: Quadratic Function g(x) = x² - 4x + 3
Graph: A parabola opening upwards with a vertex at (2, -1).
Properties: Vertex, axis of symmetry, intercepts, domain is (-∞, ∞), range is [-1, ∞).

Example 3: Exponential Function h(x) = 2^x
Graph: A curve that increases rapidly as x increases, approaching the x-axis as x decreases.
Properties: Exponential growth, horizontal asymptote at y = 0, domain is (-∞, ∞), range is (0, ∞).

Analogies & Mental Models:

Think of each type of function as a different tool in a toolbox: Each tool is designed for a specific purpose. Linear functions are like screwdrivers, quadratic functions are like wrenches, and exponential functions are like power drills.

Common Misconceptions:

❌ Students often confuse polynomial and rational functions. They might think that any function with a variable in the denominator is a rational function.
✓ Actually, a rational function must have a polynomial in both the numerator and the denominator.
Why this confusion happens: The definitions of polynomial and rational functions can be tricky to remember, and it's easy to make mistakes when identifying the function type.

Visual Description:

Imagine a gallery of different function graphs. Each graph has its own unique shape and characteristics. Linear functions are straight lines, quadratic functions are parabolas, exponential functions are curves that grow rapidly, and logarithmic functions are curves that grow slowly.

Practice Check:

Identify the type of function and describe its key characteristics:

a) f(x) = 3x - 5
b) g(x) = x² + 2x - 1
c) h(x) = 1/x

Answer:

a) Linear function: A straight line with a slope of 3 and a y-intercept of -5.
b) Quadratic function: A parabola opening upwards with a vertex at (-1, -2).
c) Rational function: Has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0.

Connection to Other Sections:

Understanding the different types of functions is essential for modeling real-world phenomena, solving equations, and analyzing data.

### 4.7 Transformations of Functions

Overview: Understanding how to transform the graph of a function is a powerful tool for visualizing and manipulating functions.

The Core Concept: Transformations of functions involve altering the graph of a function by shifting, stretching, compressing, or reflecting it. These transformations can be expressed algebraically by modifying the function's equation.

Vertical Translations: Adding a constant 'c' to the function, f(x) + c, shifts the graph vertically. If c > 0, the graph shifts upwards. If c < 0, the graph shifts downwards.
Horizontal Translations: Replacing x with (x - c) in the function, f(x - c), shifts the graph horizontally. If c > 0, the graph shifts to the right. If c < 0, the graph shifts to the left.
Vertical Stretches and Compressions: Multiplying the function by a constant 'a', a f(x), stretches or compresses the graph vertically. If a > 1, the graph stretches vertically. If 0 < a < 1, the graph compresses vertically.
*Horizontal Stretches and

Okay, I'm ready to create this comprehensive Pre-Calculus lesson on Functions. This will be a detailed and thorough resource designed to be engaging and accessible for high school students.

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## 1. INTRODUCTION

### 1.1 Hook & Context

Imagine you're developing a new social media app. You want to predict how many users you'll have in a year based on your initial marketing spend. Or, perhaps you're designing a bridge and need to understand how the weight of cars affects the bridge's structure. These scenarios, seemingly different, share a common thread: they involve understanding relationships between quantities. These relationships, at their core, are what functions are all about. We encounter functions every day, often without realizing it. The price of gasoline is a function of the cost of crude oil, the temperature you feel is a function of the actual temperature and the wind speed, and the distance a car travels is a function of its speed and the time it travels. Functions aren’t just abstract mathematical ideas; they are the language we use to describe and model the world around us.

### 1.2 Why This Matters

Functions are the bedrock of advanced mathematics, science, engineering, and countless other fields. A solid understanding of functions is essential for success in calculus, physics, computer science, economics, and more. In the real world, understanding functions allows you to analyze data, make predictions, and solve complex problems. For example, economists use functions to model economic growth, engineers use them to design structures and systems, and computer scientists use them to create algorithms. This knowledge builds directly on your algebra skills (solving equations, graphing lines) and sets the stage for understanding more advanced concepts like limits, derivatives, and integrals in calculus. This lesson isn’t just about memorizing formulas; it's about equipping you with a powerful tool for understanding and shaping the world.

### 1.3 Learning Journey Preview

In this lesson, we'll embark on a journey to explore the world of functions. We'll start by defining what a function is and how it differs from other types of relations. Then, we'll delve into different ways to represent functions – through equations, graphs, tables, and words. We'll learn how to identify key properties of functions, such as their domain, range, intercepts, and symmetry. We'll then explore different types of functions, including linear, quadratic, polynomial, rational, exponential, and logarithmic functions. We'll learn how to perform operations on functions, such as addition, subtraction, multiplication, division, and composition. Finally, we'll see how functions are used to model real-world phenomena and solve practical problems. Each concept will build on the previous one, creating a solid foundation for your future mathematical endeavors.

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## 2. LEARNING OBJECTIVES

By the end of this lesson, you will be able to:

Explain the definition of a function and distinguish it from a relation that is not a function using the vertical line test.
Determine the domain and range of a function represented graphically, algebraically, and in table form.
Identify and describe key features of functions, including intercepts, maxima, minima, intervals of increasing and decreasing behavior, and symmetry.
Graph linear, quadratic, polynomial, rational, exponential, and logarithmic functions, and identify their key characteristics.
Perform arithmetic operations (addition, subtraction, multiplication, division) and composition on functions, and determine the domain of the resulting function.
Model real-world scenarios using functions and interpret the meaning of function values, intercepts, and other key features in the context of the problem.
Analyze and compare different types of functions based on their properties, graphs, and equations.
Apply transformations (translations, reflections, stretches, and compressions) to the graphs of functions and write the corresponding equations.

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## 3. PREREQUISITE KNOWLEDGE

Before diving into functions, it's crucial to have a solid grasp of the following concepts:

Algebraic Expressions: Simplifying, factoring, and evaluating algebraic expressions.
Solving Equations: Solving linear, quadratic, and simple polynomial equations.
Graphing Lines: Understanding slope-intercept form (y = mx + b) and graphing linear equations.
Coordinate Plane: Plotting points and understanding the x and y axes.
Interval Notation: Expressing sets of numbers using interval notation (e.g., [a, b), (a, ∞)).
Set Notation: Understanding basic set operations like union and intersection.
Exponents and Radicals: Working with exponents, radicals, and their properties.

If you need a refresher on any of these topics, I recommend reviewing your algebra textbook or searching for online resources like Khan Academy. Having a strong foundation in these areas will make learning about functions much easier and more enjoyable.

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## 4. MAIN CONTENT

### 4.1 What is a Function?

Overview: At its heart, a function is a special type of relationship between two sets of values. One set is called the input (often denoted as 'x'), and the other is called the output (often denoted as 'y' or f(x)). The key characteristic of a function is that each input corresponds to exactly one output.

The Core Concept: A function is a rule that assigns each element in a set (called the domain) to exactly one element in another set (called the range). Think of a function as a machine. You put something in (the input), and the machine does something to it and spits out something else (the output). What makes it a function is that if you put the same thing in, you'll always get the same thing out. This "one-to-one" (or "many-to-one," but not "one-to-many") correspondence is the defining characteristic of a function.

Mathematically, we often write a function as y = f(x), where 'f' is the name of the function, 'x' is the input, and 'y' (or f(x)) is the output. The set of all possible inputs is called the domain of the function, and the set of all possible outputs is called the range of the function.

It's crucial to distinguish a function from a general relation. A relation is simply any set of ordered pairs (x, y). A function is a special type of relation where no two ordered pairs have the same x-value but different y-values. In other words, each x-value can only be paired with one y-value.

Concrete Examples:

Example 1: Vending Machine
Setup: A vending machine has buttons labeled A1, A2, B1, B2, etc. Each button corresponds to a specific snack or drink.
Process: You press button A1 (the input). The vending machine dispenses a bag of chips (the output).
Result: Pressing A1 will always give you the same bag of chips (assuming the machine is working correctly). This is a function because each button (input) corresponds to only one item (output).
Why this matters: This demonstrates the one-to-one correspondence. Each input has a unique output.

Example 2: Phone Number Lookup
Setup: A phone book lists names and phone numbers.
Process: You look up a person's name (the input). The phone book gives you their phone number (the output).
Result: If a person has only one phone number, this is a function. However, if a person has multiple phone numbers listed, it's not a function, because one input (the name) has multiple outputs (the phone numbers).
Why this matters: This highlights the importance of the "exactly one" condition.

Analogies & Mental Models:

Think of it like a recipe: You put in ingredients (inputs), and the recipe (function) tells you how to combine them to get a specific dish (output). If you follow the same recipe, you should get the same dish every time.
Think of it like a computer program: You give the program some data (input), and the program processes it according to its instructions (function) and produces a result (output). The same input should always produce the same output.

Common Misconceptions:

❌ Students often think that a function must be defined by a formula.
✓ Actually, a function can be defined by a formula, a graph, a table, or even a verbal description. The key is the one-to-one correspondence.
Why this confusion happens: Formulas are a common way to represent functions, but they are not the only way.

Visual Description:

Imagine a graph on the coordinate plane. Each point on the graph represents an ordered pair (x, y). For a relation to be a function, no two points can have the same x-coordinate but different y-coordinates. This leads to the Vertical Line Test: If you can draw a vertical line that intersects the graph at more than one point, then the graph does not represent a function. The vertical line represents a single x-value, and if it intersects the graph at multiple points, it means that that x-value is associated with multiple y-values, violating the definition of a function.

Practice Check:

Which of the following relations is a function?

A: {(1, 2), (2, 3), (3, 4), (4, 5)}
B: {(1, 2), (2, 3), (1, 4), (3, 5)}

Answer: A is a function because each x-value is paired with only one y-value. B is not a function because the x-value 1 is paired with both 2 and 4.

Connection to Other Sections:

This section lays the foundation for understanding all subsequent sections. Without a clear understanding of what a function is, it's impossible to grasp concepts like domain, range, and different types of functions. This leads directly into a discussion of how to represent functions.

### 4.2 Representing Functions

Overview: Functions can be represented in various ways, each providing a different perspective on the relationship between input and output. The most common representations are equations, graphs, tables, and verbal descriptions.

The Core Concept: Understanding how to represent functions in different ways is crucial for analyzing and interpreting them. Each representation has its strengths and weaknesses, and the best representation to use depends on the specific problem or situation.

Equations: An equation expresses the relationship between the input (x) and the output (y) using a mathematical formula. For example, y = 2x + 1 is an equation representing a linear function.
Graphs: A graph visually represents the function on a coordinate plane. The x-axis represents the input, and the y-axis represents the output. Each point on the graph corresponds to an ordered pair (x, y) that satisfies the function's equation.
Tables: A table lists pairs of input and output values. The table can be used to approximate the function's behavior or to represent a function when an equation is not known.
Verbal Descriptions: A verbal description explains the relationship between the input and output using words. For example, "The function squares the input and adds 3."

Concrete Examples:

Example 1: Equation to Graph
Setup: Consider the equation y = x².
Process: To graph this function, we can choose several x-values, calculate the corresponding y-values, and plot the resulting points on the coordinate plane. For example, if x = -2, y = 4; if x = -1, y = 1; if x = 0, y = 0; if x = 1, y = 1; if x = 2, y = 4.
Result: Connecting these points creates a parabola, which is the graph of the function y = x².
Why this matters: This shows how an algebraic representation (equation) can be translated into a visual representation (graph).

Example 2: Table to Equation
Setup: Consider the following table:
| x | y |
|---|---|
| 1 | 3 |
| 2 | 5 |
| 3 | 7 |
| 4 | 9 |
Process: We can observe that the y-values are increasing by 2 for each increase of 1 in the x-values. This suggests a linear relationship. We can find the slope (m) as (5-3)/(2-1) = 2. Using the point-slope form of a line (y - y1 = m(x - x1)), we can write the equation as y - 3 = 2(x - 1).
Result: Simplifying the equation, we get y = 2x + 1. This is the equation that represents the relationship shown in the table.
Why this matters: This demonstrates how to derive an algebraic representation from a tabular representation.

Analogies & Mental Models:

Think of it like different languages: Equations, graphs, tables, and verbal descriptions are like different languages that all describe the same concept (the function). Being fluent in all these "languages" allows you to understand the function from different perspectives.

Common Misconceptions:

❌ Students often think that every table represents a function.
✓ Actually, a table only represents a function if each x-value is paired with only one y-value.
Why this confusion happens: Students may not always check if the table satisfies the definition of a function.

Visual Description:

Imagine a graph of a function. The x-axis represents the input values, and the y-axis represents the output values. Each point on the graph corresponds to an ordered pair (x, y) that satisfies the function's equation. The shape of the graph provides information about the function's behavior, such as whether it is increasing or decreasing, whether it has any maximum or minimum values, and whether it is symmetric.

Practice Check:

Represent the function "The output is the square root of the input" in the following ways:

Equation: y = √x
Table: (Example)
| x | y |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 4 | 2 |
| 9 | 3 |

Connection to Other Sections:

This section builds on the previous section by showing how functions can be represented in different ways. This knowledge is essential for understanding the properties of functions and for modeling real-world phenomena. It leads into the next section on domain and range.

### 4.3 Domain and Range

Overview: The domain and range are fundamental concepts associated with functions. They define the set of possible input values and the set of corresponding output values.

The Core Concept: The domain of a function is the set of all possible input values (x-values) for which the function is defined. In other words, it's the set of all x-values that you can "plug into" the function without causing any mathematical errors (such as dividing by zero or taking the square root of a negative number).

The range of a function is the set of all possible output values (y-values or f(x)-values) that the function can produce. It's the set of all values that the function "spits out" when you plug in all possible x-values from the domain.

Determining the domain and range is crucial for understanding the behavior of a function and for interpreting its meaning in real-world applications.

Concrete Examples:

Example 1: Domain and Range of y = √x
Setup: Consider the function y = √x.
Process: The square root of a negative number is not a real number. Therefore, the input x must be greater than or equal to 0. The square root of a non-negative number is always non-negative.
Result: The domain of the function is [0, ∞), and the range is [0, ∞).
Why this matters: This shows how the mathematical definition of the function restricts the possible input and output values.

Example 2: Domain and Range of y = 1/x
Setup: Consider the function y = 1/x.
Process: Division by zero is undefined. Therefore, the input x cannot be equal to 0. As x approaches 0 from the positive side, y approaches infinity. As x approaches 0 from the negative side, y approaches negative infinity. As x approaches infinity, y approaches 0. As x approaches negative infinity, y approaches 0.
Result: The domain of the function is (-∞, 0) U (0, ∞), and the range is (-∞, 0) U (0, ∞).
Why this matters: This demonstrates how a function can have a domain and range that exclude certain values due to mathematical constraints.

Analogies & Mental Models:

Think of the domain as the "allowed" ingredients: The domain is like the list of ingredients that you're allowed to use in a recipe. If you try to use an ingredient that's not on the list, the recipe won't work.
Think of the range as the "possible" dishes: The range is like the set of dishes that you can make using the recipe. You can't make a dish that's not in the range, no matter what ingredients you use.

Common Misconceptions:

❌ Students often think that the domain is always all real numbers.
✓ Actually, the domain can be restricted by mathematical constraints (such as division by zero or square roots of negative numbers) or by the context of the problem.
Why this confusion happens: Students may not always consider the limitations imposed by the function's equation or the real-world situation.

Visual Description:

Imagine the graph of a function. The domain is the set of all x-values that are "covered" by the graph. You can visualize this by projecting the graph onto the x-axis. The range is the set of all y-values that are "covered" by the graph. You can visualize this by projecting the graph onto the y-axis.

Practice Check:

Determine the domain and range of the function represented by the following graph: (Imagine a graph of a parabola opening upwards with vertex at (0, 1))

Domain: (-∞, ∞)
Range: [1, ∞)

Connection to Other Sections:

This section builds on the previous sections by introducing the concepts of domain and range. Understanding domain and range is essential for analyzing and interpreting functions, and it leads into the next section on key features of functions.

### 4.4 Key Features of Functions

Overview: Functions possess several key features that help us understand their behavior and characteristics. These features include intercepts, maxima and minima, intervals of increasing and decreasing behavior, and symmetry.

The Core Concept: Identifying and describing these key features allows us to gain a deeper understanding of the function's properties and its relationship to the real-world phenomena it may be modeling.

Intercepts:
x-intercepts: The points where the graph of the function intersects the x-axis. At these points, y = 0.
y-intercept: The point where the graph of the function intersects the y-axis. At this point, x = 0.
Maxima and Minima:
Local Maximum: A point where the function's value is greater than or equal to the values at all nearby points.
Local Minimum: A point where the function's value is less than or equal to the values at all nearby points.
Absolute Maximum: The highest value of the function over its entire domain.
Absolute Minimum: The lowest value of the function over its entire domain.
Intervals of Increasing and Decreasing Behavior:
Increasing Interval: An interval where the function's values are increasing as x increases.
Decreasing Interval: An interval where the function's values are decreasing as x increases.
Symmetry:
Even Function: A function that is symmetric about the y-axis. Mathematically, f(-x) = f(x).
Odd Function: A function that is symmetric about the origin. Mathematically, f(-x) = -f(x).

Concrete Examples:

Example 1: Analyzing the Features of y = x² - 4
Setup: Consider the function y = x² - 4.
Process:
x-intercepts: Set y = 0: x² - 4 = 0 => x = ±2. x-intercepts are (-2, 0) and (2, 0).
y-intercept: Set x = 0: y = 0² - 4 = -4. y-intercept is (0, -4).
Minimum: The vertex of the parabola is at (0, -4), which is the absolute minimum.
Increasing/Decreasing: The function is decreasing on the interval (-∞, 0) and increasing on the interval (0, ∞).
Symmetry: The function is even because f(-x) = (-x)² - 4 = x² - 4 = f(x).
Result: We have identified the key features of the function.
Why this matters: This illustrates how to find and interpret key features from an equation.

Example 2: Analyzing Features from a Graph (Imagine a graph of a wave-like function)
Setup: Consider a graph of a periodic wave.
Process: Visually identify the x-intercepts, y-intercept, local maxima and minima, intervals of increasing and decreasing behavior, and any symmetry.
Result: Based on the graph, we can determine the approximate values of these features.
Why this matters: This demonstrates how to extract information about a function from its graphical representation.

Analogies & Mental Models:

Think of it like describing a landscape: The key features of a function are like describing the key features of a landscape – the highest peaks (maxima), the lowest valleys (minima), where the land is rising (increasing intervals), and where it is falling (decreasing intervals).

Common Misconceptions:

❌ Students often confuse local maxima/minima with absolute maxima/minima.
✓ Actually, a local maximum/minimum is only the highest/lowest value in a nearby region, while an absolute maximum/minimum is the highest/lowest value over the entire domain.
Why this confusion happens: Students may not fully understand the difference between "local" and "absolute."

Visual Description:

Imagine the graph of a function. The intercepts are the points where the graph crosses the x and y axes. The maxima and minima are the "peaks" and "valleys" of the graph. The intervals of increasing and decreasing behavior are the sections of the graph where it is going uphill or downhill. Symmetry can be seen by reflecting the graph across the y-axis (for even functions) or rotating it 180 degrees about the origin (for odd functions).

Practice Check:

Identify the intercepts, maxima/minima, and intervals of increasing/decreasing behavior for the function y = -x² + 2x + 3. (Solve this using algebraic techniques and/or a graphing calculator).

Connection to Other Sections:

This section integrates the concepts of domain, range, and representations of functions. It builds towards understanding specific types of functions.

### 4.5 Linear Functions

Overview: Linear functions are the simplest type of functions, characterized by a constant rate of change.

The Core Concept: A linear function is a function whose graph is a straight line. The general form of a linear function is y = mx + b, where 'm' is the slope (representing the rate of change) and 'b' is the y-intercept (representing the value of y when x = 0).

Linear functions are widely used to model real-world phenomena that exhibit a constant rate of change, such as distance traveled at a constant speed, the cost of items at a fixed price per item, or the depreciation of an asset over time.

Concrete Examples:

Example 1: Modeling Distance Traveled
Setup: A car travels at a constant speed of 60 miles per hour.
Process: We can model the distance traveled (y) as a function of time (x) using the equation y = 60x.
Result: After 2 hours, the car will have traveled 120 miles (y = 60 2 = 120).
Why this matters: This shows how a linear function can be used to model a simple real-world scenario.

Example 2: Finding the Equation of a Line
Setup: A line passes through the points (1, 3) and (2, 5).
Process: The slope of the line is (5 - 3) / (2 - 1) = 2. Using the point-slope form of a line (y - y1 = m(x - x1)), we can write the equation as y - 3 = 2(x - 1).
Result: Simplifying the equation, we get y = 2x + 1.
Why this matters: This demonstrates how to find the equation of a linear function given two points on the line.

Analogies & Mental Models:

Think of it like a ramp: A linear function is like a ramp with a constant slope. The steeper the ramp, the greater the slope of the function.

Common Misconceptions:

❌ Students often confuse slope with y-intercept.
✓ Actually, the slope represents the rate of change, while the y-intercept represents the initial value.
Why this confusion happens: Students may not fully understand the meaning of each parameter in the linear equation.

Visual Description:

The graph of a linear function is a straight line. The slope determines the steepness of the line, and the y-intercept determines where the line crosses the y-axis. A positive slope indicates an increasing function, while a negative slope indicates a decreasing function. A slope of zero indicates a horizontal line.

Practice Check:

Find the equation of the line that is parallel to y = 3x - 2 and passes through the point (0, 5).

Connection to Other Sections:

This section introduces a specific type of function, linear functions. This builds a foundation for understanding more complex function types.

### 4.6 Quadratic Functions

Overview: Quadratic functions are another important type of function, characterized by a parabolic shape.

The Core Concept: A quadratic function is a function that can be written in the form y = ax² + bx + c, where 'a', 'b', and 'c' are constants and 'a' is not equal to 0. The graph of a quadratic function is a parabola.

The key features of a quadratic function include its vertex (the maximum or minimum point of the parabola), its x-intercepts (the points where the parabola crosses the x-axis), and its y-intercept (the point where the parabola crosses the y-axis).

Quadratic functions are used to model various real-world phenomena, such as the trajectory of a projectile, the shape of a suspension bridge cable, or the relationship between price and demand.

Concrete Examples:

Example 1: Modeling Projectile Motion
Setup: A ball is thrown upwards with an initial velocity of 48 feet per second from a height of 6 feet.
Process: We can model the height of the ball (y) as a function of time (x) using the equation y = -16x² + 48x + 6 (where -16 represents half the acceleration due to gravity).
Result: We can find the maximum height of the ball by finding the vertex of the parabola. The x-coordinate of the vertex is -b / (2a) = -48 / (2 -16) = 1.5. The y-coordinate of the vertex is -16 (1.5)² + 48 1.5 + 6 = 42. The maximum height of the ball is 42 feet.
Why this matters: This shows how a quadratic function can be used to model a real-world problem involving projectile motion.

Example 2: Finding the Vertex of a Parabola
Setup: Consider the quadratic function y = 2x² - 8x + 5.
Process: The x-coordinate of the vertex is -b / (2a) = -(-8) / (2 2) = 2. The y-coordinate of the vertex is 2 (2)² - 8 2 + 5 = -3.
Result: The vertex of the parabola is (2, -3).
Why this matters: This demonstrates how to find the vertex of a parabola using the formula -b / (2a).

Analogies & Mental Models:

Think of it like a water fountain: The water shoots up (increasing) and then falls back down (decreasing), forming a parabolic arc. The highest point is the vertex.

Common Misconceptions:

❌ Students often think that all parabolas open upwards.
✓ Actually, a parabola opens upwards if the coefficient 'a' is positive and downwards if the coefficient 'a' is negative.
Why this confusion happens: Students may not pay attention to the sign of the leading coefficient.

Visual Description:

The graph of a quadratic function is a parabola. The vertex is the highest or lowest point on the parabola. The x-intercepts are the points where the parabola crosses the x-axis. The y-intercept is the point where the parabola crosses the y-axis. If 'a' is positive, the parabola opens upwards. If 'a' is negative, the parabola opens downwards.

Practice Check:

Find the vertex, x-intercepts, and y-intercept of the quadratic function y = x² + 4x + 3.

Connection to Other Sections:

This section builds on the previous section by introducing another specific type of function, quadratic functions.

### 4.7 Polynomial Functions

Overview: Polynomial functions are a broad class of functions that include linear and quadratic functions as special cases.

The Core Concept: A polynomial function is a function that can be written in the form y = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀, where 'a_n', 'a_n-1', ..., 'a₁', 'a₀' are constants and 'n' is a non-negative integer (the degree of the polynomial).

The key features of polynomial functions include their degree, their leading coefficient, their end behavior, and their zeros (x-intercepts).

Polynomial functions are used to model a wide variety of real-world phenomena, such as population growth, economic trends, and the shape of curves.

Concrete Examples:

Example 1: Analyzing End Behavior
Setup: Consider the polynomial function y = x³ - 2x² + x - 1.
Process: The degree of the polynomial is 3 (odd), and the leading coefficient is 1 (positive). Therefore, as x approaches infinity, y approaches infinity, and as x approaches negative infinity, y approaches negative infinity.
Result: The end behavior of the function is that it rises to the right and falls to the left.
Why this matters: This demonstrates how to determine the end behavior of a polynomial function based on its degree and leading coefficient.

Example 2: Finding Zeros
Setup: Consider the polynomial function y = (x - 1)(x + 2)(x - 3).
Process: The zeros of the function are the values of x that make y equal to 0. Setting each factor equal to 0, we get x - 1 = 0 => x = 1, x + 2 = 0 => x = -2, and x - 3 = 0 => x = 3.
Result: The zeros of the function are 1, -2, and 3.
Why this matters: This demonstrates how to find the zeros of a polynomial function by factoring.

Analogies & Mental Models:

Think of it like a roller coaster: Polynomial functions can have multiple "hills" and "valleys," representing local maxima and minima. The degree of the polynomial determines how many "turns" the roller coaster can have.

Common Misconceptions:

❌ Students often think that the degree of a polynomial is the highest power of x that appears in the function.
✓ Actually, the degree of a polynomial is the highest power of x after the function has been simplified.
Why this confusion happens: Students may not fully simplify the function before identifying the degree.

Visual Description:

The graph of a polynomial function can have multiple "turns" and "wiggles." The degree of the polynomial determines the maximum number of turns. The leading coefficient determines the end behavior of the function. The zeros are the points where the graph crosses the x-axis.

Practice Check:

Determine the degree, leading coefficient, end behavior, and zeros of the polynomial function y = 2x⁴ - 3x² + 1.

Connection to Other Sections:

This section generalizes the concepts from linear and quadratic functions to polynomial functions.

### 4.8 Rational Functions

Overview: Rational functions are functions that are formed by dividing two polynomials.

The Core Concept: A rational function is a function that can be written in the form y = P(x) / Q(x), where P(x) and Q(x) are polynomial functions and Q(x) is not equal to 0.

The key features of rational functions include their domain (which excludes the values of x that make the denominator equal to 0), their vertical asymptotes (the vertical lines where the function approaches infinity or negative infinity), their horizontal asymptotes (the horizontal lines that the function approaches as x approaches infinity or negative infinity), and their intercepts.

Rational functions are used to model various real-world phenomena, such as the concentration of a drug in the bloodstream over time, the relationship between supply and demand, and the behavior of electrical circuits.

Concrete Examples:

Example 1: Finding Vertical Asymptotes
Setup: Consider the rational function y = 1 / (x - 2).
Process: The denominator is equal to 0 when x = 2. Therefore, there is a vertical asymptote at x = 2.
Result: The function approaches infinity as x approaches 2 from the right and approaches negative infinity as x approaches 2 from the left.
Why this matters: This demonstrates how to find vertical asymptotes by finding the values of x that make the denominator equal to 0.

Example 2: Finding Horizontal Asymptotes
Setup: Consider the rational function y = (2x + 1) / (

Okay, here is a comprehensive pre-calculus lesson on functions, designed to be engaging, thorough, and accessible for high school students. This is a substantial lesson, aiming for the specified depth and detail.

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## 1. INTRODUCTION

### 1.1 Hook & Context

Imagine you're designing a roller coaster. The height of the coaster at any point along the track depends on the distance traveled from the start. Or, think about your phone bill. The total cost depends on how many gigabytes of data you use. Even something as simple as the temperature outside changes throughout the day. What do these scenarios have in common? They all involve relationships where one thing depends on another. These types of relationships are the foundation of functions, one of the most powerful and versatile concepts in mathematics. Functions aren't just abstract formulas; they are the language we use to describe and model the world around us. We'll explore how to define, analyze, and apply functions to understand and predict these real-world phenomena.

### 1.2 Why This Matters

Functions are the bedrock upon which much of higher-level mathematics and science is built. In calculus, you'll use functions to model rates of change and accumulation. In physics, functions describe motion, forces, and energy. In computer science, functions are the building blocks of algorithms and programs. Engineers of all types (civil, electrical, mechanical, etc.) use functions to design structures, circuits, and machines. Even fields like economics and finance rely heavily on functions to model market trends and investment strategies. Understanding functions provides you with a powerful toolkit for solving problems and making informed decisions in a variety of disciplines. This lesson builds upon your prior knowledge of equations and graphs, and it sets the stage for future topics like trigonometry, exponential functions, and calculus. Mastering functions now will give you a significant advantage in your future studies and career endeavors.

### 1.3 Learning Journey Preview

In this lesson, we will embark on a journey to explore the world of functions. We'll start by defining what a function is and how it differs from other types of relations. Then, we'll learn how to represent functions in various ways: equations, graphs, tables, and verbal descriptions. We'll delve into the key properties of functions, such as domain, range, intercepts, and symmetry. We'll also explore different types of functions, including linear, quadratic, polynomial, rational, and piecewise functions. Finally, we'll learn how to manipulate functions through transformations and combinations, and we'll see how functions are used to model real-world phenomena. Each concept will build upon the previous one, providing you with a solid foundation in the theory and application of functions.

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## 2. LEARNING OBJECTIVES

By the end of this lesson, you will be able to:

1. Define a function and distinguish it from a relation, explaining the vertical line test.
2. Represent functions using equations, graphs, tables, and verbal descriptions, and translate between these representations.
3. Determine the domain and range of a function given its equation, graph, or table.
4. Identify and interpret key features of a function's graph, including intercepts, intervals of increase/decrease, maxima/minima, and symmetry.
5. Classify functions into common types (linear, quadratic, polynomial, rational, piecewise) and recognize their characteristic properties.
6. Perform operations on functions, including addition, subtraction, multiplication, division, and composition.
7. Apply transformations (translations, reflections, stretches, compressions) to the graph of a function and write the corresponding equation.
8. Model real-world scenarios using functions and interpret the results in context.

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## 3. PREREQUISITE KNOWLEDGE

Before diving into functions, you should have a solid understanding of the following concepts:

Basic Algebra: Solving equations and inequalities, simplifying expressions, working with exponents and radicals.
Coordinate Plane: Plotting points, understanding the x and y axes, identifying quadrants.
Linear Equations: Slope-intercept form, point-slope form, finding the equation of a line given two points or a point and a slope.
Graphing: Plotting points to create graphs of simple equations.
Sets and Set Notation: Understanding what a set is and how to represent it using roster notation and set-builder notation.
Interval Notation: Representing intervals of numbers using parentheses and brackets.

Review Resources: If you need a refresher on any of these topics, you can consult your algebra textbook, online resources like Khan Academy, or previous class notes. Pay particular attention to the sections covering linear equations, graphing, and solving equations.

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## 4. MAIN CONTENT

### 4.1 What is a Function?

Overview: At its core, a function is a special type of relationship between two sets of elements. It's a rule that assigns each element from one set (the input) to exactly one element in another set (the output). This "one-to-one" or "many-to-one" nature is what distinguishes a function from other types of relations.

The Core Concept: A function is a relation between a set of inputs, called the domain, and a set of permissible outputs, called the range, with the property that each input is related to exactly one output. Think of a function like a vending machine. You put in money (the input), and you get out a specific snack or drink (the output). You wouldn't expect to put in the same amount of money and get two different items at the same time. That's the essence of a function: for each input, there's only one possible output.

We often use the notation f(x) to represent a function, where x is the input and f(x) is the output. f(x) is read as "f of x". For instance, if f(x) = x², then f(3) = 3² = 9. This means that when the input is 3, the output is 9. The variable 'x' is often referred to as the independent variable, and f(x) is the dependent variable because its value depends on the value of x.

It's crucial to distinguish a function from a general relation. A relation is simply any set of ordered pairs. A function is a special kind of relation where no two ordered pairs have the same first element (input) but different second elements (outputs). If a relation has two different y-values for the same x-value, it's not a function.

Concrete Examples:

Example 1: The Square Function
Setup: Consider the function f(x) = x².
Process: For any real number x, the function squares it. For example, f(2) = 2² = 4, f(-2) = (-2)² = 4, f(0) = 0² = 0.
Result: Each input x produces a unique output x². Notice that different inputs can produce the same output (e.g., 2 and -2 both map to 4), but one input never produces two different outputs.
Why this matters: This is a fundamental example of a function. It demonstrates that different inputs can lead to the same output, but one input must have only one output.

Example 2: Relation that is NOT a Function
Setup: Consider the relation defined by x = y².
Process: If we choose x = 4, then y² = 4, which means y = 2 or y = -2.
Result: The input x = 4 produces two different outputs, y = 2 and y = -2.
Why this matters: This violates the definition of a function. Because one input yields two possible outputs, x = y² defines a relation, but not a function.

Analogies & Mental Models:

Think of it like... a machine that takes in raw materials (the input) and produces a finished product (the output). Each set of raw materials will only produce one final product.
How the analogy maps: The raw materials are the 'x' values (domain), and the finished product is the 'f(x)' value (range). The machine itself is the rule or equation that defines the function.
Where the analogy breaks down: A machine can break down or be faulty, producing inconsistent results. A function, by definition, is always consistent.

Common Misconceptions:

❌ Students often think that a function cannot have the same output for different inputs.
✓ Actually, a function can have the same output for different inputs, as long as each input has only one output. For instance, f(x) = x² is a function even though f(2) = f(-2) = 4.
Why this confusion happens: The focus is often on the one-to-one aspect, but the crucial part is that each input must have one output.

Visual Description:

Imagine a graph on the coordinate plane. To visually determine if a graph represents a function, we use the Vertical Line Test. This test states that if any vertical line drawn on the graph intersects the graph at more than one point, then the graph does not represent a function. The vertical line represents a single x-value, and the points of intersection represent the corresponding y-values. If a vertical line intersects the graph at two or more points, it means that one x-value has multiple y-values, violating the definition of a function.

Practice Check:

Question: Does the equation y = ±√x represent a function? Why or why not?

Answer: No, y = ±√x does not represent a function. For any positive value of x, there are two corresponding values of y (one positive and one negative). For example, if x = 4, then y = ±2, meaning one input (4) has two outputs (2 and -2). This violates the definition of a function.

Connection to Other Sections: This section is foundational. All subsequent sections rely on a clear understanding of what a function is. The next section will explore how to represent functions in different ways.

### 4.2 Representing Functions

Overview: Functions can be expressed in several different ways, each offering unique insights and advantages. The most common representations are equations, graphs, tables, and verbal descriptions. Being able to translate between these representations is a crucial skill for understanding and working with functions.

The Core Concept: A function can be described using an equation that defines the relationship between the input and output. For example, f(x) = 2x + 1 is an equation that represents a function. A graph is a visual representation of the function, showing the relationship between the input and output as points on the coordinate plane. A table lists pairs of input and output values. A verbal description explains the function in words. Each representation provides a different perspective on the function, and being able to move seamlessly between them allows for a deeper understanding.

Concrete Examples:

Example 1: Linear Function
Equation: f(x) = 3x - 2
Graph: A straight line with a slope of 3 and a y-intercept of -2. Plot points like (0, -2), (1, 1), (2, 4) and connect them.
Table:

| x | f(x) |
| ---- | ---- |
| -1 | -5 |
| 0 | -2 |
| 1 | 1 |
| 2 | 4 |

Verbal Description: "This function takes an input, multiplies it by 3, and then subtracts 2 to get the output."

Example 2: Quadratic Function
Equation: g(x) = x² - 4
Graph: A parabola that opens upwards, with its vertex at (0, -4) and x-intercepts at -2 and 2.
Table:

| x | g(x) |
| ---- | ---- |
| -3 | 5 |
| -2 | 0 |
| -1 | -3 |
| 0 | -4 |
| 1 | -3 |
| 2 | 0 |
| 3 | 5 |

Verbal Description: "This function takes an input, squares it, and then subtracts 4 to get the output."

Analogies & Mental Models:

Think of it like... different languages describing the same idea. The equation is like a mathematical language, the graph is like a visual language, the table is like a data-driven language, and the verbal description is like an everyday language.
How the analogy maps: Each language conveys the same information about the function, but in a different format.
Where the analogy breaks down: The mathematical language (equation) is often the most precise and concise, while the everyday language (verbal description) can be more accessible but less precise.

Common Misconceptions:

❌ Students often think that a table can completely define a function.
✓ Actually, a table only provides a finite number of data points. It doesn't necessarily tell you what happens between those points. The equation or graph provides a more complete picture.
Why this confusion happens: Tables are easy to understand, but they can be misleading if you don't know the underlying equation or pattern.

Visual Description:

Imagine a function as a mapping diagram. The domain (input values) is represented by a set of points on the left, and the range (output values) is represented by a set of points on the right. Arrows connect each input to its corresponding output. The graph is a visual representation of these arrows plotted on the coordinate plane. The equation is the symbolic representation of the mapping rule.

Practice Check:

Question: Given the verbal description "The function doubles the input and adds 5," write the equation, create a table with three input-output pairs, and sketch a rough graph of the function.

Answer:
Equation: f(x) = 2x + 5
Table:

| x | f(x) |
| ---- | ---- |
| -1 | 3 |
| 0 | 5 |
| 1 | 7 |

Graph: A straight line passing through the points (-1, 3), (0, 5), and (1, 7).

Connection to Other Sections: This section builds on the previous section by showing how to represent functions once we understand what they are. The next section will focus on the key properties of functions that can be identified from these representations.

### 4.3 Domain and Range

Overview: The domain and range are fundamental properties of a function. The domain specifies all possible input values for which the function is defined, while the range specifies all possible output values that the function can produce. Understanding domain and range is essential for interpreting and applying functions correctly.

The Core Concept: The domain of a function is the set of all possible input values (x-values) for which the function is defined. In other words, it's the set of all x-values that you can plug into the function without causing any mathematical errors (like dividing by zero or taking the square root of a negative number). The range of a function is the set of all possible output values (f(x)-values or y-values) that the function can produce when you plug in all the values from the domain. Visually, the domain is the set of all x-values covered by the graph of the function, and the range is the set of all y-values covered by the graph.

Concrete Examples:

Example 1: Linear Function
Equation: f(x) = 2x + 1
Domain: All real numbers. There are no restrictions on the input values. We can plug in any real number for x and get a valid output. In interval notation, this is written as (-∞, ∞).
Range: All real numbers. The function can produce any real number as an output. In interval notation, this is written as (-∞, ∞).

Example 2: Square Root Function
Equation: g(x) = √x
Domain: x ≥ 0. We cannot take the square root of a negative number. In interval notation, this is written as [0, ∞).
Range: g(x) ≥ 0. The square root of a non-negative number is always non-negative. In interval notation, this is written as [0, ∞).

Example 3: Rational Function
Equation: h(x) = 1/(x - 2)
Domain: All real numbers except x = 2. We cannot divide by zero, so x - 2 cannot be equal to zero. Therefore, x ≠ 2. In interval notation, this is written as (-∞, 2) ∪ (2, ∞).
Range: All real numbers except y = 0. The function can produce any real number as an output except 0. In interval notation, this is written as (-∞, 0) ∪ (0, ∞).

Analogies & Mental Models:

Think of it like... a machine with specific input requirements. The domain is like the set of materials that the machine can process without breaking down. The range is like the set of products that the machine can produce.
How the analogy maps: The machine can only accept certain inputs (domain) and can only produce certain outputs (range).
Where the analogy breaks down: The machine analogy doesn't fully capture the mathematical precision and consistency of functions.

Common Misconceptions:

❌ Students often forget to consider restrictions on the domain, especially when dealing with square roots or fractions.
✓ Always check for values that would cause division by zero or taking the square root of a negative number.
Why this confusion happens: It's easy to focus on the equation itself and forget to think about the potential restrictions on the input values.

Visual Description:

To find the domain from a graph, imagine shining a light from the top and bottom of the graph onto the x-axis. The shadow cast on the x-axis represents the domain. Similarly, to find the range, imagine shining a light from the left and right sides of the graph onto the y-axis. The shadow cast on the y-axis represents the range.

Practice Check:

Question: Determine the domain and range of the function represented by the graph of a parabola that opens upwards, with its vertex at (1, -3).

Answer:
Domain: All real numbers, or (-∞, ∞). The parabola extends infinitely to the left and right.
Range: y ≥ -3, or [-3, ∞). The lowest point on the parabola is at y = -3, and it extends upwards infinitely.

Connection to Other Sections: Understanding domain and range is crucial for interpreting the behavior of functions. The next section will explore other key features of functions, such as intercepts, symmetry, and intervals of increase/decrease, which are all related to the domain and range.

### 4.4 Key Features of Function Graphs

Overview: The graph of a function is a rich source of information. By analyzing the graph, we can identify key features such as intercepts, intervals of increase/decrease, maxima/minima, and symmetry, which provide insights into the function's behavior.

The Core Concept: Intercepts are the points where the graph of the function intersects the x-axis (x-intercepts) or the y-axis (y-intercept). X-intercepts are also called roots or zeros of the function. Intervals of increase/decrease describe where the function's values are increasing or decreasing as x increases. A maximum is the highest point on the graph within a certain interval, and a minimum is the lowest point. Symmetry describes how the graph is reflected or rotated. A function is even if its graph is symmetric about the y-axis, and it's odd if its graph is symmetric about the origin.

Concrete Examples:

Example 1: Quadratic Function
Equation: f(x) = x² - 4
Graph: Parabola opening upwards.
X-intercepts: (-2, 0) and (2, 0). These are the points where the graph crosses the x-axis. They are found by setting f(x) = 0 and solving for x.
Y-intercept: (0, -4). This is the point where the graph crosses the y-axis. It is found by setting x = 0 and evaluating f(0).
Intervals of Decrease: (-∞, 0). The function is decreasing as x increases from negative infinity to 0.
Intervals of Increase: (0, ∞). The function is increasing as x increases from 0 to positive infinity.
Minimum: (0, -4). This is the vertex of the parabola, the lowest point on the graph.
Symmetry: Even function. The graph is symmetric about the y-axis. f(x) = f(-x) for all x.

Example 2: Linear Function
Equation: f(x) = -2x + 3
Graph: A straight line with a negative slope.
X-intercept: (1.5, 0).
Y-intercept: (0, 3).
Intervals of Decrease: (-∞, ∞). The function is decreasing for all values of x.
Intervals of Increase: None.
Maximum/Minimum: None. The function increases and decreases indefinitely.
Symmetry: Neither even nor odd.

Analogies & Mental Models:

Think of it like... reading a map. The graph is like a map of the function's behavior. The intercepts are like landmarks, the intervals of increase/decrease are like hills and valleys, and the maxima/minima are like the highest and lowest points on the terrain.
How the analogy maps: Each feature of the graph provides information about the function's characteristics, just like landmarks and terrain features provide information about a geographical area.
Where the analogy breaks down: A map is a static representation of a physical space, while a function's graph represents a dynamic relationship between variables.

Common Misconceptions:

❌ Students often confuse maxima and minima with the endpoints of the domain.
✓ Maxima and minima are the highest and lowest points within a certain interval of the domain. They are not necessarily the endpoints of the domain.
Why this confusion happens: Students may focus on the overall shape of the graph and not pay attention to the local behavior.

Visual Description:

Imagine walking along the graph of a function from left to right. If you are going uphill, the function is increasing. If you are going downhill, the function is decreasing. The points where you change direction (from uphill to downhill or vice versa) are the maxima and minima.

Practice Check:

Question: Describe the key features of the graph of a cubic function that has x-intercepts at -1, 0, and 2, a local maximum between -1 and 0, and a local minimum between 0 and 2.

Answer:
X-intercepts: (-1, 0), (0, 0), (2, 0)
Y-intercept: (0, 0)
Intervals of Increase: (-∞, x₁) and (x₂, ∞), where x₁ is the x-coordinate of the local maximum and x₂ is the x-coordinate of the local minimum.
Intervals of Decrease: (x₁, x₂)
Local Maximum: At some point between -1 and 0.
Local Minimum: At some point between 0 and 2.
Symmetry: Likely neither even nor odd.

Connection to Other Sections: Understanding these key features allows us to analyze and compare different types of functions. The next section will classify functions into common types based on their equations and graphs.

### 4.5 Types of Functions

Overview: Functions can be classified into different types based on their equations and graphs. Recognizing these types allows you to quickly understand their key properties and behavior. Common types include linear, quadratic, polynomial, rational, and piecewise functions.

The Core Concept: A linear function has the form f(x) = mx + b, where m is the slope and b is the y-intercept. Its graph is a straight line. A quadratic function has the form f(x) = ax² + bx + c, where a ≠ 0. Its graph is a parabola. A polynomial function has the form f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where n is a non-negative integer. A rational function is a function that can be written as the ratio of two polynomial functions, f(x) = p(x)/q(x), where q(x) ≠ 0. A piecewise function is defined by different equations over different intervals of its domain.

Concrete Examples:

Example 1: Linear Function
Equation: f(x) = 2x - 3
Graph: A straight line with slope 2 and y-intercept -3.
Key Properties: Constant rate of change (slope), domain and range are all real numbers (unless restricted).

Example 2: Quadratic Function
Equation: f(x) = x² + 2x - 1
Graph: A parabola.
Key Properties: Vertex, axis of symmetry, maximum or minimum value, x-intercepts (roots).

Example 3: Polynomial Function
Equation: f(x) = x³ - 3x² + 2x
Graph: A curve with turning points.
Key Properties: Degree, leading coefficient, end behavior, x-intercepts.

Example 4: Rational Function
Equation: f(x) = 1/(x - 1)
Graph: A hyperbola with vertical and horizontal asymptotes.
Key Properties: Vertical asymptotes (where the denominator is zero), horizontal asymptotes (determined by the degrees of the numerator and denominator), domain restrictions.

Example 5: Piecewise Function
Equation:

``
f(x) = {
x + 1, if x < 0
x², if x ≥ 0
}
``

Graph: A combination of a line and a parabola.
Key Properties: Defined by different equations over different intervals, may have discontinuities (jumps) at the boundaries of the intervals.

Analogies & Mental Models:

Think of it like... different types of vehicles. A linear function is like a bicycle, a quadratic function is like a car, a polynomial function is like a truck, a rational function is like a motorcycle with a sidecar (asymptotes), and a piecewise function is like a transformer that can change into different vehicles depending on the situation.
How the analogy maps: Each type of vehicle has different characteristics and capabilities, just like each type of function has different properties and behaviors.
Where the analogy breaks down: This analogy is more about visualizing the different types, not about the mathematical properties themselves.

Common Misconceptions:

❌ Students often think that all polynomial functions have the same end behavior.
✓ The end behavior of a polynomial function depends on its degree and leading coefficient.
Why this confusion happens: Students may focus on specific examples and not generalize the rules for end behavior.

Visual Description:

Visualize the characteristic shape of each function type: a straight line for linear, a parabola for quadratic, a smooth curve with turning points for polynomial, a hyperbola with asymptotes for rational, and a combination of different shapes for piecewise.

Practice Check:

Question: Identify the type of function represented by the equation f(x) = (x² + 1)/(x - 2). Describe its key properties.

Answer: This is a rational function. Its key properties include a vertical asymptote at x = 2, a horizontal asymptote at y = 1 (since the degrees of the numerator and denominator are the same), and a domain of all real numbers except x = 2.

Connection to Other Sections: Classifying functions into types allows us to apply specific techniques for analyzing and manipulating them. The next section will explore operations on functions, such as addition, subtraction, multiplication, division, and composition.

### 4.6 Operations on Functions

Overview: Just like numbers, functions can be combined using arithmetic operations to create new functions. These operations include addition, subtraction, multiplication, division, and composition. Understanding these operations allows you to build more complex functions from simpler ones.

The Core Concept: Given two functions f(x) and g(x), we can define the following operations:

Addition: (f + g)(x) = f(x) + g(x)
Subtraction: (f - g)(x) = f(x) - g(x)
Multiplication: (f g)(x) = f(x) g(x)
Division: (f / g)(x) = f(x) / g(x), where g(x) ≠ 0
Composition: (f ∘ g)(x) = f(g(x)). This means we first evaluate g(x) and then use the result as the input for f(x).

Concrete Examples:

Example 1: Addition and Subtraction
f(x) = x² + 1
g(x) = 2x - 3
(f + g)(x) = (x² + 1) + (2x - 3) = x² + 2x - 2
(f - g)(x) = (x² + 1) - (2x - 3) = x² - 2x + 4

Example 2: Multiplication and Division
f(x) = x + 2
g(x) = x - 1
(f g)(x) = (x + 2)(x - 1) = x² + x - 2
(f / g)(x) = (x + 2) / (x - 1), where x ≠ 1

Example 3: Composition
f(x) = √x
g(x) = x + 2
(f ∘ g)(x) = f(g(x)) = f(x + 2) = √(x + 2)
(g ∘ f)(x) = g(f(x)) = g(√x) = √x + 2

Analogies & Mental Models:

Think of it like... a recipe. The functions are like ingredients, and the operations are like cooking methods. Adding, subtracting, multiplying, and dividing functions are like mixing ingredients together in different proportions. Composition is like following a sequence of steps to combine ingredients in a specific order.
How the analogy maps: Each operation produces a new dish (function) with different properties, just like each cooking method produces a different dish with different flavors.
Where the analogy breaks down: This analogy doesn't fully capture the mathematical precision and consistency of function operations.

Common Misconceptions:

❌ Students often confuse the order of operations in function composition.
✓ Remember that (f ∘ g)(x) = f(g(x)), meaning you evaluate g(x) first and then use the result as the input for f(x).
Why this confusion happens: The notation can be confusing, and it's easy to mix up the order of the functions.

Visual Description:

Imagine two machines, f and g. For function composition f(g(x)), the output of machine g becomes the input of machine f. You are essentially chaining the machines together.

Practice Check:

Question: Given f(x) = x² - 1 and g(x) = 3x, find (f ∘ g)(x) and (g ∘ f)(x).

Answer:
(f ∘ g)(x) = f(g(x)) = f(3x) = (3x)² - 1 = 9x² - 1
(g ∘ f)(x) = g(f(x)) = g(x² - 1) = 3(x² - 1) = 3x² - 3

Connection to Other Sections: Understanding operations on functions allows us to transform and manipulate functions in various ways. The next section will explore transformations of functions, such as translations, reflections, stretches, and compressions.

### 4.7 Transformations of Functions

Overview: Transformations of functions involve altering the graph of a function by shifting it, reflecting it, stretching it, or compressing it. Understanding these transformations allows you to quickly sketch the graph of a transformed function based on the graph of the original function.

The Core Concept: Given a function f(x), we can apply the following transformations:

Vertical Translation: f(x) + c shifts the graph c units upward if c > 0 and c units downward if c < 0.
Horizontal Translation: f(x - c) shifts the graph c units to the right if c > 0 and c units to the left if c < 0.
Vertical Stretch/Compression: a f(x) stretches the graph vertically by a factor of a if a > 1 and compresses it vertically by a factor of a if 0 < a < 1. If a < 0, it also reflects the graph across the x-axis.
Horizontal Stretch/Compression: *f(