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{\pi}{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 $g(x)$, we can determine if they are inverses of each other by verifying if their composition results in the original input. 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?

Before we dive into the verification process, let's briefly discuss what inverse functions are. An inverse function is a function that reverses the operation of another function. In other words, if we have a function $f(x)$, its inverse function $g(x)$ will take the output of $f(x)$ and return the original input. Mathematically, this can be represented as:

$f(g(x)) = x$

Verifying Inverse Functions

To verify that $g(x)=\frac{1}{3}x$ is the inverse of $f(x)=3x$, we need to check if their composition results in the original input. We can do this by substituting $g(x)$ into $f(x)$ and simplifying the expression.

Step 1: Substitute $g(x)$ into $f(x)$

We will substitute $g(x)=\frac{1}{3}x$ into $f(x)=3x$.

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

Step 2: Simplify the Expression

Now, we will simplify the expression by multiplying $3$ and $\frac{1}{3}$.

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

Conclusion

As we can see, the composition of $f(x)$ and $g(x)$ results in the original input $x$. This confirms that $g(x)=\frac{1}{3}x$ is indeed the inverse of $f(x)=3x$.

Answer

Based on our verification process, we can conclude that the correct expression to verify that $g(x)$ is the inverse of $f(x)$ is:

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

Discussion

In this article, we explored how to verify that $g(x)=\frac{1}{3}x$ is the inverse of $f(x)=3x$. We discussed the concept of inverse functions, the process of verifying inverse functions, and provided a step-by-step guide on how to simplify the expression. By following these steps, we can confidently determine if two functions are inverses of each other.

Common Mistakes

When verifying inverse functions, it's essential to avoid common mistakes. Here are a few:

• Not simplifying the expression: Failing to simplify the expression can lead to incorrect conclusions.
• Not checking the composition: Not verifying the composition of the two functions can result in incorrect conclusions.
• Not using the correct notation: Using the wrong notation can lead to confusion and incorrect conclusions.

Conclusion

In conclusion, verifying inverse functions is a crucial step in understanding the relationship between different functions. By following the steps outlined in this article, we can confidently determine if two functions are inverses of each other. Remember to simplify the expression, check the composition, and use the correct notation to avoid common mistakes.

Final Answer

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Introduction

In our previous article, we explored how to verify that $g(x)=\frac{1}{3}x$ is the inverse of $f(x)=3x$. 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 we have a function $f(x)$, its inverse function $g(x)$ will take the output of $f(x)$ and return the original input.

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

A: To find the inverse of a function, we need to swap the x and y variables and then solve for y. For example, if we have a function $f(x) = 2x + 3$, we can find its inverse by swapping the x and y variables and then solving for y.

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

A: The relationship between a function and its inverse is that they are "reversible". In other words, if we have a function $f(x)$ and its inverse $g(x)$, we can use them to "undo" each other.

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 determined by the function itself.

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, we need to check if their composition results in the original input. We can do this by substituting the second function into the first function and simplifying the expression.

Q: What is the importance of inverse functions in real-world applications?

A: Inverse functions are used in many real-world applications, such as:

• Physics: Inverse functions are used to describe the relationship between different physical quantities, such as distance and velocity.
• Engineering: Inverse functions are used to design and optimize systems, such as electronic circuits and mechanical systems.
• Computer Science: Inverse functions are used in algorithms and data structures, such as sorting and searching.

Q: Can I use inverse functions to solve equations?

A: Yes, inverse functions can be used to solve equations. By using the inverse of a function, we can "undo" the function and solve for the input variable.

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 simplifying the expression: Failing to simplify the expression can lead to incorrect conclusions.
• Not checking the composition: Not verifying the composition of the two functions can result in incorrect conclusions.
• Not using the correct notation: Using the wrong notation can lead to confusion and incorrect conclusions.

Conclusion

In conclusion, inverse functions are an essential concept in mathematics and have many real-world applications. By understanding how to find and verify inverse functions, we can solve equations and optimize systems. Remember to avoid common mistakes and use the correct notation to ensure accurate results.

Final Answer

The final answer is that inverse functions are a powerful tool for solving equations and optimizing systems. By understanding how to find and verify inverse functions, we can unlock new possibilities and solve complex problems.