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

Introduction

In mathematics, inverse functions play a crucial role in solving equations and understanding the relationships 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 $f^{-1}(f(x)) = x$. This means that if we apply the inverse function to the output of the original function, we get back the original input.

Verifying Inverse Functions

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 finding $f(g(x))$.

Finding $f(g(x))$

To find $f(g(x))$, we need to substitute $g(x) = \frac{1}{3}x$ into the function $f(x) = 3x$. This gives us:

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

Simplifying this 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 $f(x)$.

Finding $g(f(x))$

To find $g(f(x))$, we need to substitute $f(x) = 3x$ into the function $g(x) = \frac{1}{3}x$. This gives us:

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

Simplifying this 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 $f(x) = 3x$ by verifying that $f(g(x)) = x$ and $g(f(x)) = x$. This means that if we apply the inverse function to the output of the original function, we get back the original input.

Answer

The correct answer is A. $3x\left(\frac{x}{3}\right)$.

Discussion

This problem requires a deep understanding of inverse functions and how to verify them. It also requires the ability to simplify expressions and manipulate functions. In this article, we have shown that $g(x) = \frac{1}{3}x$ is the inverse of $f(x) = 3x$ by verifying that $f(g(x)) = x$ and $g(f(x)) = x$. This is a fundamental concept in mathematics and is used extensively in various fields, including physics, engineering, and economics.

Example Use Cases

Inverse functions have many practical applications in various fields. Here are a few examples:

• In physics, inverse functions are used to describe the relationship between different physical quantities, such as velocity and time.
• In engineering, inverse functions are used to design and optimize systems, such as control systems and signal processing systems.
• In economics, inverse functions are used to model the relationship between different economic variables, such as supply and demand.

Conclusion

$$$$$$$$

Introduction

Inverse functions are a fundamental concept in mathematics that have many practical applications in various fields. 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&A

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, then its inverse function, denoted as $f^{-1}(x)$, satisfies the condition $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

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 are some common types of inverse functions?

A: Some common types of inverse functions include:

• Linear inverse functions: These are inverse functions that are linear in form, such as $f(x) = mx + b$ and $f^{-1}(x) = \frac{x - b}{m}$.
• Quadratic inverse functions: These are inverse functions that are quadratic in form, such as $f(x) = ax^2 + bx + c$ and $f^{-1}(x) = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• Exponential inverse functions: These are inverse functions that are exponential in form, such as $f(x) = a^x$ and $f^{-1}(x) = \log_a(x)$.

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 need to show that the composition of the two functions is equal to the identity function. In other words, you need to show that $f(g(x)) = x$ and $g(f(x)) = x$.

Q: What are some real-world applications of inverse functions?

A: Inverse functions have many practical applications in various fields, including:

• Physics: Inverse functions are used to describe the relationship between different physical quantities, such as velocity and time.
• Engineering: Inverse functions are used to design and optimize systems, such as control systems and signal processing systems.
• Economics: Inverse functions are used to model the relationship between different economic variables, such as supply and demand.

Q: Can you provide some examples of inverse functions?

A: Here are a few examples of inverse functions:

• $f(x) = 2x$ and $f^{-1}(x) = \frac{x}{2}$
• $f(x) = x^2$ and $f^{-1}(x) = \sqrt{x}$
• $f(x) = e^x$ and $f^{-1}(x) = \ln(x)$

Conclusion

In conclusion, inverse functions are a fundamental concept in mathematics that have many practical applications in various fields. By understanding how to find and verify inverse functions, you can solve a wide range of problems in physics, engineering, economics, and other fields. We hope that this Q&A guide has been helpful in answering your questions about inverse functions.

Additional Resources

If you are interested in learning more about inverse functions, here are some additional resources that you may find helpful:

• Khan Academy: Inverse Functions
• Mathway: Inverse Functions
• Wolfram Alpha: Inverse Functions

Final Thoughts

Inverse functions are a powerful tool for solving problems in mathematics and other fields. By understanding how to find and verify inverse functions, you can unlock a wide range of possibilities for solving problems and modeling real-world phenomena. We hope that this Q&A guide has been helpful in answering your questions about inverse functions.