# 35 Inverses Of Linear Functions Common Core Algebra 2 Homework

## Introduction

Linear functions and their inverses are fundamental concepts in algebra. In this article, we will explore the topic of inverses of linear functions, specifically in the context of Common Core Algebra 2 homework. Understanding the concept of inverses is crucial for solving equations, finding the domain and range of functions, and analyzing the behavior of functions. Whether you're a student struggling with your homework or a parent trying to help your child, this article will provide you with a comprehensive guide to mastering inverses of linear functions.

## What are Linear Functions?

Before diving into inverses, it's important to have a clear understanding of linear functions. A linear function is a function that can be represented by a straight line on a graph. It follows the form f(x) = mx + b, where m represents the slope of the line and b represents the y-intercept.

### The Slope of a Linear Function

The slope of a linear function determines how steep the line is. It represents the rate of change between the x and y values. A positive slope indicates an upward slope, while a negative slope indicates a downward slope. A slope of zero indicates a horizontal line, and an undefined slope indicates a vertical line.

### The Y-Intercept of a Linear Function

The y-intercept is the point where the line intersects the y-axis. It represents the value of y when x is equal to zero. The y-intercept is often denoted as (0, b), where b is the y-coordinate.

## What are Inverse Functions?

An inverse function undoes the action of the original function. In the context of linear functions, finding the inverse involves swapping the x and y variables. The inverse function of f(x) = mx + b can be represented as f^{-1}(x) = (x - b) / m.

### Properties of Inverse Functions

There are a few key properties to keep in mind when working with inverse functions:

- The domain of the original function becomes the range of the inverse function.
- The range of the original function becomes the domain of the inverse function.
- The composition of a function and its inverse results in the identity function.

## Finding the Inverse of a Linear Function

To find the inverse of a linear function, follow these steps:

- Replace f(x) with y in the original function.
- Swap the x and y variables to isolate y on one side of the equation.
- Re-write the equation in the form y = mx + b.
- Replace y with f
^{-1}(x) to represent the inverse function.

### Example

Let's find the inverse of the linear function f(x) = 2x + 3:

- Replace f(x) with y: y = 2x + 3.
- Swap the x and y variables: x = 2y + 3.
- Isolate y: x - 3 = 2y.
- Divide by 2: (x - 3) / 2 = y.
- Replace y with f
^{-1}(x): f^{-1}(x) = (x - 3) / 2.

## Verifying Inverse Functions

Once you have found the inverse of a linear function, you can verify its correctness by checking if the composition of the function and its inverse results in the identity function. This means that when you plug the inverse function into the original function, you should get back the input value.

### Example

Let's verify the inverse function we found earlier, f^{-1}(x) = (x - 3) / 2, by composing it with the original function f(x) = 2x + 3:

- Compose the functions: f(f
^{-1}(x)) = f((x - 3) / 2). - Simplify the expression: f((x - 3) / 2) = 2((x - 3) / 2) + 3.
- Distribute and simplify further: 2((x - 3) / 2) + 3 = (x - 3) + 3 = x.

As we can see, the composition of the function and its inverse results in the identity function x, verifying that the inverse function is correct.

## Using Inverse Functions to Solve Equations

In addition to verifying the correctness of inverse functions, they can also be used to solve equations involving linear functions. By applying the inverse function to both sides of the equation, you can isolate the variable and find its value.

### Example

Let's use the inverse function we found earlier, f^{-1}(x) = (x - 3) / 2, to solve the equation f(x) = 5:

- Replace f(x) with 5: 5 = 2x + 3.
- Subtract 3 from both sides: 5 - 3 = 2x.
- Divide by 2: 2 = x.

Therefore, the solution to the equation f(x) = 5 is x = 2.

## Finding the Domain and Range of Inverse Functions

When working with inverse functions, it's important to determine their domain and range. The domain of the inverse function corresponds to the range of the original function, and vice versa.

### Domain of the Inverse Function

The domain of the inverse function is determined by the range of the original function. It represents the set of all possible input values for which the inverse function is defined. To find the domain of the inverse function, consider the domain of the original function.

### Range of the Inverse Function

The range of the inverse function is determined by the domain of the original function. It represents the set of all possible output values for the inverse function. To find the range of the inverse function, consider the range of the original function.

## Conclusion

Inverses of linear functions are a fundamental concept in algebra. They allow us to undo the action of a linear function and provide valuable insights into the behavior of functions. By understanding the properties of inverse functions, finding the inverse of a linear function, verifying its correctness, and using it to solve equations, you can confidently tackle Common Core Algebra 2 homework problems involving inverses of linear functions. Remember to consider the domain and range of inverse functions to ensure their validity. With practice and perseverance, you'll become proficient in handling inverses and open the door to more advanced mathematical concepts.