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

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 it satisfies the condition $f(g(x)) = x$ and $g(f(x)) = x$. In this article, we will explore how to verify if $g(x)$ is the inverse of $f(x)$ using a specific expression.

Understanding the Given Functions

We are given two functions:

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

Our goal is to verify if $g(x)$ is the inverse of $f(x)$.

The Concept of Inverse Functions

To understand the concept of inverse functions, let's consider the following:

• If $f(x)$ is a function, then its inverse function, $g(x)$, is denoted as $f^{-1}(x)$.
• The inverse function $g(x)$ satisfies the condition $f(g(x)) = x$.
• Similarly, the original function $f(x)$ satisfies the condition $g(f(x)) = x$.

Verifying the Inverse Function

To verify if $g(x)$ is the inverse of $f(x)$, we need to check if the composition of $f(x)$ and $g(x)$ satisfies the condition $f(g(x)) = x$.

Let's start by finding the composition of $f(x)$ and $g(x)$:

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

Substituting the value of $g(x)$ into the function $f(x)$, we get:

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

Simplifying the expression, we get:

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

$f(g(x)) = x$

This shows that the composition of $f(x)$ and $g(x)$ satisfies the condition $f(g(x)) = x$, which means that $g(x)$ is indeed the inverse of $f(x)$.

Evaluating the Given Expressions

Now, let's evaluate the given expressions to verify if they can be used to verify $g(x)$ is the inverse of $f(x)$.

A. $\frac{1}{5}\left(\frac{1}{5}x + 5\right) + 5$

This expression is not the correct composition of $f(x)$ and $g(x)$. It is simply a rearrangement of the terms in the expression for $g(x)$.

B. $\frac{1}{5}(5x - 25) + 5$

This expression is also not the correct composition of $f(x)$ and $g(x)$. It is simply a rearrangement of the terms in the expression for $f(x)$.

Conclusion

In conclusion, to verify if $g(x)$ is the inverse of $f(x)$, we need to check if the composition of $f(x)$ and $g(x)$ satisfies the condition $f(g(x)) = x$. We have shown that the composition of $f(x)$ and $g(x)$ indeed satisfies this condition, which means that $g(x)$ is indeed the inverse of $f(x)$.

Therefore, the correct expression to verify $g(x)$ is the inverse of $f(x)$ is not among the given options A and B.

Final Answer

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Introduction

In our previous article, we explored how to find the inverse of a function and verified if $g(x)$ is the inverse of $f(x)$. In this article, we will answer some frequently asked questions related to finding the inverse of a function.

Q: What is the inverse of a function?

A: The inverse of a function is a function that undoes the action of the original function. In other words, if $f(x)$ is a function, then its inverse function, $g(x)$, satisfies the condition $f(g(x)) = x$.

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

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

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

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

A: A function and its inverse are two different functions that are related to each other. The function $f(x)$ and its inverse $g(x)$ satisfy the condition $f(g(x)) = x$ and $g(f(x)) = x$.

Q: Can a function have more than one inverse?

A: No, a function cannot have more than one inverse. The inverse of a function is unique and is denoted as $f^{-1}(x)$.

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

A: To verify if a function is the inverse of another function, you need to check if the composition of the two functions satisfies the condition $f(g(x)) = x$.

Q: What is the importance of finding the inverse of a function?

A: Finding the inverse of a function is important in many areas of mathematics, such as calculus, algebra, and geometry. It helps us to solve equations, find the domain and range of a function, and understand the behavior of a function.

Q: Can I use a calculator to find the inverse of a function?

A: Yes, you can use a calculator to find the inverse of a function. However, it's always a good idea to verify the result by checking if the composition of the two functions satisfies the condition $f(g(x)) = x$.

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

A: To graph the inverse of a function, you need to follow these steps:

1. Graph the original function.
2. Reflect the graph of the original function across the line $y = x$.

Conclusion

In conclusion, finding the inverse of a function is an important concept in mathematics that has many applications. By following the steps outlined in this article, you can find the inverse of a function and verify if it is the inverse of another function.

Final Answer

The final answer is: None

Additional Resources

• Inverse Functions
• Finding the Inverse of a Function
• Graphing the Inverse of a Function