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.

Introduction

In mathematics, inverse functions play a crucial role in solving equations and understanding the relationship between different 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 that $g(x) = \frac{1}{3}x$ is the inverse of $f(x) = 3x$.

What are Inverse Functions?

An inverse function is a function that reverses the operation of another function. In other words, if $f(x)$ is a function, then its inverse function, denoted as $f^{-1}(x)$, satisfies the condition: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$. In the context of this problem, we are looking for a function $g(x)$ that satisfies the condition: $f(g(x)) = x$ and $g(f(x)) = x$.

Verifying the Inverse Function

To verify that $g(x) = \frac{1}{3}x$ is the inverse of $f(x) = 3x$, we need to show that $f(g(x)) = x$ and $g(f(x)) = x$. Let's start by evaluating $f(g(x))$.

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

Using the definition of $f(x)$, we get:

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

Simplifying the expression, we get:

$f(g(x)) = x$

This shows that $f(g(x)) = x$, which is one of the conditions for $g(x)$ to be the inverse of $f(x)$.

Evaluating the Options

Now, let's evaluate the options given to us:

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

This expression is not equal to $x$, so it cannot be used to verify that $g(x)$ is the inverse of $f(x)$.

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

This expression is equal to $x$, which is one of the conditions for $g(x)$ to be the inverse of $f(x)$. However, we need to show that $g(f(x)) = x$ as well.

Let's evaluate $g(f(x))$:

$g(f(x)) = g(3x)$

Using the definition of $g(x)$, we get:

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

Simplifying the expression, we get:

$g(f(x)) = x$

This shows that $g(f(x)) = x$, which is the other condition for $g(x)$ to be the inverse of $f(x)$.

Conclusion

In conclusion, we have shown that $g(x) = \frac{1}{3}x$ is the inverse of $f(x) = 3x$. We have also evaluated the options given to us and found that option B is the correct expression to verify that $g(x)$ is the inverse of $f(x)$.

Final Answer

The final answer is:

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Introduction

In our previous article, we explored the concept of inverse functions and how to verify that a given function is the inverse of another function. In this article, we will answer some frequently asked questions about inverse functions.

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

A: A function and its inverse are two functions that "undo" each other. In other words, if $f(x)$ is a function, then its inverse function, denoted as $f^{-1}(x)$, satisfies the condition: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$. This means that if you apply the function and then its inverse, you will get back the original input.

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. For example, if you have a function $f(x) = 2x + 3$, you can find its inverse by swapping the x and y variables and then solving for y:

$y = 2x + 3$

$x = 2y + 3$

Solving for y, you get:

$y = \frac{x - 3}{2}$

This is the inverse of the original function.

Q: What is the domain and range of an inverse function?

A: The domain and range of an inverse function are the same as the range and domain of the original function, respectively. In other words, if the original function has a domain of $[a, b]$ and a range of $[c, d]$, then the inverse function will have a domain of $[c, d]$ and a range of $[a, b]$.

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 $f^{-1}(x)$. If you have two different functions that satisfy the condition: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$, then they are the same function.

Q: How do I graph an inverse function?

A: To graph an inverse function, you can use the following steps:

1. Graph the original function.
2. Reflect the graph of the original function across the line y = x.
3. The resulting graph is the graph of the inverse function.

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

A: Some common mistakes to avoid when working with inverse functions include:

• Not checking if the function is one-to-one before finding its inverse.
• Not swapping the x and y variables when finding the inverse of a function.
• Not solving for y when finding the inverse of a function.
• Not checking if the inverse function is a function.

Conclusion

In conclusion, inverse functions are an important concept in mathematics that can be used to solve equations and understand the relationship between different functions. By understanding the properties and behavior of inverse functions, you can use them to solve a wide range of problems.

Final Tips

• Make sure to check if the function is one-to-one before finding its inverse.
• Always swap the x and y variables when finding the inverse of a function.
• Solve for y when finding the inverse of a function.
• Check if the inverse function is a function.

By following these tips and understanding the properties and behavior of inverse functions, you can use them to solve a wide range of problems and become a more confident and proficient mathematician.