If $f(x) = 5x$, What Is $f^{-1}(x$\]?A. $f^{-1}(x) = -5x$ B. $f^{-1}(x) = \frac{1}{5}$ C. $f^{-1}(x) = \frac{1}{5}x$ D. $f^{-1}(x) = 5x$

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Introduction

Inverse functions are a crucial concept in mathematics, particularly in algebra and calculus. They play a vital role in solving equations, graphing functions, and understanding the behavior of functions. In this article, we will delve into the concept of inverse functions, explore the process of finding the inverse of a function, and apply this knowledge to solve for $f^{-1}(x)$ in the given function $f(x) = 5x$.

What is an Inverse Function?

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 $f^{-1}(x)$ will take the output of $f(x)$ and return the original input. This means that if $f(x) = y$, then $f^{-1}(y) = x$.

Properties of Inverse Functions

Inverse functions have several important properties that make them useful in mathematics. Some of these properties include:

• One-to-One Correspondence: An inverse function is a one-to-one correspondence between the input and output values of the original function.
• Symmetry: The graph of an inverse function is symmetric to the graph of the original function with respect to the line $y = x$.
• Reversibility: The inverse function reverses the operation of the original function.

Finding the Inverse of a Function

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

1. Replace $f(x)$ with $y$: Replace the function $f(x)$ with $y$ to simplify the notation.
2. Interchange $x$ and $y$: Interchange the $x$ and $y$ variables to get $x = f(y)$.
3. Solve for $y$: Solve the resulting equation for $y$ to get the inverse function $f^{-1}(x)$.

Solving for $f^{-1}(x)$ in $f(x) = 5x$

Now, let's apply the steps above to solve for $f^{-1}(x)$ in the given function $f(x) = 5x$.

Step 1: Replace $f(x)$ with $y$

Replace $f(x)$ with $y$ to get $y = 5x$.

Step 2: Interchange $x$ and $y$

Interchange the $x$ and $y$ variables to get $x = 5y$.

Step 3: Solve for $y$

Solve the resulting equation for $y$ to get $y = \frac{x}{5}$.

Step 4: Replace $y$ with $f^{-1}(x)$

Replace $y$ with $f^{-1}(x)$ to get $f^{-1}(x) = \frac{x}{5}$.

Conclusion

In this article, we explored the concept of inverse functions, their properties, and the process of finding the inverse of a function. We applied this knowledge to solve for $f^{-1}(x)$ in the given function $f(x) = 5x$. The inverse function $f^{-1}(x) = \frac{x}{5}$ is the correct answer.

Final Answer

The final answer is $\boxed{f^{-1}(x) = \frac{x}{5}}$.

References

• [1] "Inverse Functions" by Khan Academy
• [2] "Inverse Functions" by Math Open Reference
• [3] "Inverse Functions" by Wolfram MathWorld
  Inverse Functions: A Comprehensive Q&A Guide =====================================================

Introduction

Inverse functions are a fundamental concept in mathematics, particularly in algebra and calculus. In our previous article, we explored the concept of inverse functions, their properties, and the process of finding the inverse of a function. In this article, we will provide a comprehensive Q&A guide to help you understand inverse functions better.

Q: What is an inverse function?

A: 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 $f^{-1}(x)$ will take the output of $f(x)$ and return the original input.

Q: What are the properties of inverse functions?

A: Inverse functions have several important properties that make them useful in mathematics. Some of these properties include:

• One-to-One Correspondence: An inverse function is a one-to-one correspondence between the input and output values of the original function.
• Symmetry: The graph of an inverse function is symmetric to the graph of the original function with respect to the line $y = x$.
• Reversibility: The inverse function reverses the operation of the original function.

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 $y$: Replace the function $f(x)$ with $y$ to simplify the notation.
2. Interchange $x$ and $y$: Interchange the $x$ and $y$ variables to get $x = f(y)$.
3. Solve for $y$: Solve the resulting equation for $y$ to get the inverse function $f^{-1}(x)$.

Q: Can you provide an example of finding the inverse of a function?

A: Let's consider the function $f(x) = 5x$. To find the inverse of this function, we need to follow the steps above.

Step 1: Replace $f(x)$ with $y$

Replace $f(x)$ with $y$ to get $y = 5x$.

Step 2: Interchange $x$ and $y$

Interchange the $x$ and $y$ variables to get $x = 5y$.

Step 3: Solve for $y$

Solve the resulting equation for $y$ to get $y = \frac{x}{5}$.

Step 4: Replace $y$ with $f^{-1}(x)$

Replace $y$ with $f^{-1}(x)$ to get $f^{-1}(x) = \frac{x}{5}$.

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 $f^{-1}(x)$ are two different functions that are symmetric to each other with respect to the line $y = x$.

Q: Can you provide an example of a function and its inverse?

A: Let's consider the function $f(x) = 2x$. The inverse of this function is $f^{-1}(x) = \frac{x}{2}$. The graph of $f(x)$ and $f^{-1}(x)$ are symmetric to each other with respect to the line $y = x$.

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

A: Inverse functions have numerous applications in real-life situations, such as:

• Physics: Inverse functions are used to describe the relationship between variables in physics, such as distance, velocity, and acceleration.
• 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.

Conclusion

In this article, we provided a comprehensive Q&A guide to help you understand inverse functions better. We covered the concept of inverse functions, their properties, and the process of finding the inverse of a function. We also provided examples and real-life applications of inverse functions.

Final Answer

The final answer is $\boxed{f^{-1}(x) = \frac{x}{5}}$.

References

• [1] "Inverse Functions" by Khan Academy
• [2] "Inverse Functions" by Math Open Reference
• [3] "Inverse Functions" by Wolfram MathWorld