If F ( X ) = 3 X F(x) = 3x F ( X ) = 3 X And G ( X ) = 1 3 X G(x) = \frac{1}{3}x G ( X ) = 3 1 X , Which Expression Could Be Used To Verify That G ( X G(x G ( X ] Is The Inverse Of F ( X F(x F ( X ]?A. 3x\left(\frac{x}{3}\right ] B. ( 1 3 X ) ( 3 X \left(\frac{1}{3}x\right)(3x ( 3 1 X ) ( 3 X ] C.

Mar 10, 2025 by ADMIN 305 views

**Understanding Inverse Functions: A Step-by-Step Guide**

Introduction

In mathematics, inverse functions play a crucial role in solving equations and analyzing the behavior of functions. Given two functions, $f(x)$ and $g(x)$ , the inverse of $f(x)$ is denoted as $f^{-1}(x)$ and is defined as a function that undoes the action of $f(x)$ . In other words, if $f(x)$ maps an input $x$ to an output $y$ , then the inverse function $f^{-1}(x)$ maps the output $y$ back to the input $x$ . In this article, we will explore how to verify that a given function is the inverse of another function using a specific expression.

The Given Functions

We are given two functions:

$f(x) = 3x$
$g(x) = \frac{1}{3}x$

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

What is an Inverse Function?

An inverse function is a function that undoes the action of another function. In other words, if $f(x)$ maps an input $x$ to an output $y$ , then the inverse function $f^{-1}(x)$ maps the output $y$ back to the input $x$ . To verify that a given function is the inverse of another function, we need to check if the composition of the two functions is equal to the identity function.

The Composition of Functions

The composition of two functions $f(x)$ and $g(x)$ is defined as:

$(f \circ g)(x) = f(g(x))$

In our case, we need to check if the composition of $f(x)$ and $g(x)$ is equal to the identity function.

Verifying the Inverse Function

To verify that $g(x)$ is the inverse of $f(x)$ , we need to check if the composition of $f(x)$ and $g(x)$ is equal to the identity function. We can do this by plugging in the expression for $g(x)$ into the expression for $f(x)$ and simplifying.

Let's start by plugging in the expression for $g(x)$ into the expression for $f(x)$ :

$f(g(x)) = 3\left(\frac{1}{3}x\right)$

Simplifying this expression, we get:

$f(g(x)) = x$

This shows that the composition of $f(x)$ and $g(x)$ is equal to the identity function, which means that $g(x)$ is indeed the inverse of $f(x)$ .

Conclusion

In this article, we explored how to verify that a given function is the inverse of another function using a specific expression. We used the given functions $f(x) = 3x$ and $g(x) = \frac{1}{3}x$ to demonstrate how to check if the composition of the two functions is equal to the identity function. By following these steps, we can verify that $g(x)$ is indeed the inverse of $f(x)$ .

The Correct Answer

Based on our analysis, the correct answer is:

B. $\left(\frac{1}{3}x\right)(3x)$

This expression represents the composition of $f(x)$ and $g(x)$ , which is equal to the identity function. Therefore, $g(x)$ is indeed the inverse of $f(x)$ .

Additional Examples

To further illustrate the concept of inverse functions, let's consider a few more examples.

Example 1

Suppose we have two functions:

$f(x) = 2x$
$g(x) = \frac{1}{2}x$

Can we verify that $g(x)$ is the inverse of $f(x)$ ?

To do this, we need to check if the composition of $f(x)$ and $g(x)$ is equal to the identity function. Let's start by plugging in the expression for $g(x)$ into the expression for $f(x)$ :

$f(g(x)) = 2\left(\frac{1}{2}x\right)$

Simplifying this expression, we get:

$f(g(x)) = x$

This shows that the composition of $f(x)$ and $g(x)$ is equal to the identity function, which means that $g(x)$ is indeed the inverse of $f(x)$ .

Example 2

Suppose we have two functions:

$f(x) = x^2$
$g(x) = \sqrt{x}$

Can we verify that $g(x)$ is the inverse of $f(x)$ ?

To do this, we need to check if the composition of $f(x)$ and $g(x)$ is equal to the identity function. Let's start by plugging in the expression for $g(x)$ into the expression for $f(x)$ :

$f(g(x)) = (x^2)^2$

Simplifying this expression, we get:

$f(g(x)) = x^4$

This shows that the composition of $f(x)$ and $g(x)$ is not equal to the identity function, which means that $g(x)$ is not the inverse of $f(x)$ .

Example 3

Suppose we have two functions:

$f(x) = \sin(x)$
$g(x) = \arcsin(x)$

Can we verify that $g(x)$ is the inverse of $f(x)$ ?

To do this, we need to check if the composition of $f(x)$ and $g(x)$ is equal to the identity function. Let's start by plugging in the expression for $g(x)$ into the expression for $f(x)$ :

$f(g(x)) = \sin(\arcsin(x))$

Simplifying this expression, we get:

$f(g(x)) = x$

This shows that the composition of $f(x)$ and $g(x)$ is equal to the identity function, which means that $g(x)$ is indeed the inverse of $f(x)$ .

Conclusion

Introduction

Inverse functions are a fundamental concept in mathematics, and understanding them is crucial for solving equations and analyzing the behavior of functions. In our previous article, we explored how to verify that a given function is the inverse of another function 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 undoes the action of another function. In other words, if $f(x)$ maps an input $x$ to an output $y$ , then the inverse function $f^{-1}(x)$ maps the output $y$ back to the input $x$ .

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

A: To find the inverse of a function, you need to swap the $x$ and $y$ variables and then solve for $y$ . This will give you the inverse function.

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 maps an input $x$ to an output $y$ , while the inverse function maps the output $y$ back to the input $x$ .

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

A: To verify that a given function is the inverse of another function, you need to check if the composition of the two functions is equal to the identity function. This can be done by plugging in the expression for the given function into the expression for the other function and simplifying.

Q: What is the composition of two functions?

A: The composition of two functions $f(x)$ and $g(x)$ is defined as:

$(f \circ g)(x) = f(g(x))$

This means that you need to plug in the expression for $g(x)$ into the expression for $f(x)$ and simplify.

Q: How do I know if a function is one-to-one?

A: A function is one-to-one if it passes the horizontal line test. This means that for every $x$ value, there is only one $y$ value. If a function is one-to-one, then it has an inverse.

Q: What is the significance of inverse functions in real-life applications?

A: Inverse functions have many real-life applications, such as:

Physics: Inverse functions are used to describe the motion of objects under the influence of forces.
Engineering: Inverse functions are used to design and analyze electrical circuits.
Computer Science: Inverse functions are used in algorithms for solving equations and analyzing the behavior of functions.

Q: Can you provide some examples of inverse functions?

A: Yes, here are some examples of inverse functions:

Linear Functions: If $f(x) = 2x$ , then the inverse function is $f^{-1}(x) = \frac{x}{2}$ .
Quadratic Functions: If $f(x) = x^2$ , then the inverse function is $f^{-1}(x) = \sqrt{x}$ .
Trigonometric Functions: If $f(x) = \sin(x)$ , then the inverse function is $f^{-1}(x) = \arcsin(x)$ .

Conclusion

In conclusion, inverse functions are a fundamental concept in mathematics that have many real-life applications. By understanding how to find and verify inverse functions, you can solve equations and analyze the behavior of functions more effectively. We hope that this Q&A article has helped you better understand inverse functions and their significance in mathematics and real-life applications.