If $f(x) = 5x - 25$ And $g(x) = \frac{1}{5}x + 5$, Which Expression Could Be Used To Verify That $g(x$\] Is The Inverse Of $f(x$\]?A. $\frac{1}{5}\left(5x - 25\right) + 5$B. $5\left(\frac{1}{5}x + 5\right)

Introduction

In mathematics, the concept of inverse functions is crucial in understanding the relationship between two functions. Given two functions, $f(x)$ and $g(x)$, if $g(x)$ is the inverse of $f(x)$, then the composition of $f(x)$ and $g(x)$ should result in the input value $x$. In this article, we will explore how to verify that $g(x)$ is the inverse of $f(x)$ using a specific expression.

Understanding Inverse Functions

Before we dive into the problem, let's understand what an inverse function is. An inverse function is a function that reverses the operation of another function. In other words, if $f(x)$ is a function that takes an input $x$ and produces an output $y$, then the inverse function $g(x)$ takes the output $y$ and produces the input $x$.

The Given Functions

We are given two functions:

• $f(x) = 5x - 25$
• $g(x) = \frac{1}{5}x + 5$

Our goal is to verify that $g(x)$ is the inverse of $f(x)$ using a specific expression.

Verifying the Inverse Function

To verify that $g(x)$ is the inverse of $f(x)$, we need to show that the composition of $f(x)$ and $g(x)$ results in the input value $x$. In other words, we need to show that $f(g(x)) = x$.

Let's start by substituting $g(x)$ into $f(x)$:

$f(g(x)) = 5\left(\frac{1}{5}x + 5\right) - 25$

Now, let's simplify the expression:

$f(g(x)) = x + 25 - 25$

$f(g(x)) = x$

As we can see, the composition of $f(x)$ and $g(x)$ results in the input value $x$. This confirms that $g(x)$ is indeed the inverse of $f(x)$.

Conclusion

In this article, we explored how to verify that $g(x)$ is the inverse of $f(x)$ using a specific expression. We started by understanding the concept of inverse functions and then applied this concept to the given functions $f(x) = 5x - 25$ and $g(x) = \frac{1}{5}x + 5$. By substituting $g(x)$ into $f(x)$ and simplifying the expression, we confirmed that $g(x)$ is indeed the inverse of $f(x)$.

The Correct Answer

Based on our analysis, the correct answer is:

• B. $5\left(\frac{1}{5}x + 5\right)$

This expression represents the composition of $f(x)$ and $g(x)$, which results in the input value $x$. Therefore, it can be used to verify that $g(x)$ is the inverse of $f(x)$.

Additional Tips and Tricks

• When working with inverse functions, it's essential to remember that the composition of a function and its inverse should result in the input value $x$.
• To verify that $g(x)$ is the inverse of $f(x)$, you can substitute $g(x)$ into $f(x)$ and simplify the expression.
• Make sure to check your work carefully to ensure that the composition of $f(x)$ and $g(x)$ results in the input value $x$.

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Introduction

In our previous article, we explored how to verify that $g(x)$ is the inverse of $f(x)$ using a specific expression. In this article, we will answer some frequently asked questions about inverse functions to help you better understand this concept.

Q: What is an inverse function?

A: An inverse function is a function that reverses the operation of another function. In other words, if $f(x)$ is a function that takes an input $x$ and produces an output $y$, then the inverse function $g(x)$ takes the output $y$ and produces the input $x$.

Q: How do I find the inverse of a function?

A: To find the inverse of a function, you can follow these steps:

1. Replace $f(x)$ with $y$.
2. Swap $x$ and $y$.
3. Solve for $y$.

For example, if we have the function $f(x) = 2x + 3$, we can find its inverse by following these steps:

1. Replace $f(x)$ with $y$: $y = 2x + 3$
2. Swap $x$ and $y$: $x = 2y + 3$
3. Solve for $y$: $y = \frac{x - 3}{2}$

Therefore, the inverse of $f(x) = 2x + 3$ is $f^{-1}(x) = \frac{x - 3}{2}$.

Q: What is the difference between a function and its inverse?

A: A function and its inverse are two different functions that work together to produce the same output. The function takes an input $x$ and produces an output $y$, while the inverse function takes the output $y$ and produces the input $x$.

For example, if we have the function $f(x) = 2x + 3$, its inverse is $f^{-1}(x) = \frac{x - 3}{2}$. When we input $x$ into $f(x)$, we get $y = 2x + 3$. Then, when we input $y$ into $f^{-1}(x)$, we get $x = \frac{y - 3}{2}$.

Q: How do I verify that a function is the inverse of another function?

A: To verify that a function is the inverse of another function, you can follow these steps:

1. Substitute the function into the inverse function.
2. Simplify the expression.
3. Check if the result is equal to the input value $x$.

For example, if we want to verify that $g(x) = \frac{1}{5}x + 5$ is the inverse of $f(x) = 5x - 25$, we can follow these steps:

1. Substitute $g(x)$ into $f(x)$: $f(g(x)) = 5\left(\frac{1}{5}x + 5\right) - 25$
2. Simplify the expression: $f(g(x)) = x + 25 - 25$
3. Check if the result is equal to the input value $x$: $f(g(x)) = x$

Therefore, we can confirm that $g(x) = \frac{1}{5}x + 5$ is indeed the inverse of $f(x) = 5x - 25$.

Q: What are some common mistakes to avoid when working with inverse functions?

A: Here are some common mistakes to avoid when working with inverse functions:

• Not checking if the function is one-to-one (injective).
• Not checking if the function is onto (surjective).
• Not verifying that the function is the inverse of another function.
• Not simplifying the expression when verifying that a function is the inverse of another function.

By avoiding these common mistakes, you can ensure that you are working with inverse functions correctly.

Conclusion

In this article, we answered some frequently asked questions about inverse functions to help you better understand this concept. We covered topics such as finding the inverse of a function, verifying that a function is the inverse of another function, and common mistakes to avoid when working with inverse functions. By following these tips and tricks, you can confidently work with inverse functions and apply them to real-world problems.