If F ( X ) = 5 X F(x) = 5x F ( X ) = 5 X , What Is F − 1 ( X F^{-1}(x F − 1 ( X ]?A. F − 1 ( X ) = − 5 X F^{-1}(x) = -5x F − 1 ( X ) = − 5 X B. F − 1 ( X ) = − 1 5 X F^{-1}(x) = -\frac{1}{5}x F − 1 ( X ) = − 5 1 ​ X C. F − 1 ( X ) = 1 5 X F^{-1}(x) = \frac{1}{5}x F − 1 ( X ) = 5 1 ​ X D. F − 1 ( X ) = 5 X F^{-1}(x) = 5x F − 1 ( X ) = 5 X

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 the original 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 $f(a) = b$, then $f^{-1}(b) = a$.

Finding the Inverse of a Function

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

1. Replace $f(x)$ with $y$: We start by replacing the function $f(x)$ with $y$. This gives us the equation $y = f(x)$.
2. Interchange $x$ and $y$: We then interchange the variables $x$ and $y$, which gives us the equation $x = f(y)$.
3. Solve for $y$: Finally, we solve for $y$ in terms of $x$. This will give us the inverse function $f^{-1}(x)$.

Finding the Inverse of $f(x) = 5x$

Now that we have a good understanding of how to find the inverse of a function, let's apply this knowledge to the given function $f(x) = 5x$.

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

We start by replacing the function $f(x)$ with $y$, which gives us the equation $y = 5x$.

Step 2: Interchange $x$ and $y$

Next, we interchange the variables $x$ and $y$, which gives us the equation $x = 5y$.

Step 3: Solve for $y$

Finally, we solve for $y$ in terms of $x$. To do this, we divide both sides of the equation by 5, which gives us:

$$y = \frac{x}{5}$$

Therefore, the inverse function $f^{-1}(x)$ is given by:

$$f^{-1}(x) = \frac{x}{5}$$

Conclusion

In this article, we explored the concept of inverse functions, learned how to find the inverse of a function, and applied this knowledge to solve for $f^{-1}(x)$ in the given function $f(x) = 5x$. We found that the inverse function $f^{-1}(x)$ is given by $f^{-1}(x) = \frac{x}{5}$. This result is consistent with option C, which states that $f^{-1}(x) = \frac{1}{5}x$. However, it's essential to note that the correct answer is $f^{-1}(x) = \frac{x}{5}$, not $f^{-1}(x) = \frac{1}{5}x$.

Final Answer

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

References

• [1] "Inverse Functions" by Khan Academy
• [2] "Inverse Functions" by Math Open Reference
• [3] "Inverse Functions" by Wolfram MathWorld
  Inverse Functions: A 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, learned how to find the inverse of a function, and applied this knowledge to solve for $f^{-1}(x)$ in the given function $f(x) = 5x$. In this article, we will continue to delve into the world of inverse functions by answering some frequently asked questions.

Q&A

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

A: A function and its inverse are two different functions that "undo" each other. 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: How do I know if a function has an inverse?

A: A function has an inverse if it is one-to-one, meaning that each output value corresponds to exactly one input value. In other words, if $f(x) = f(y)$, then $x = y$.

Q: What is the notation for the inverse of a function?

A: The notation for the inverse of a function is $f^{-1}(x)$. This is read as "f inverse of x".

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 $y$: We start by replacing the function $f(x)$ with $y$. This gives us the equation $y = f(x)$.
2. Interchange $x$ and $y$: We then interchange the variables $x$ and $y$, which gives us the equation $x = f(y)$.
3. Solve for $y$: Finally, we solve for $y$ in terms of $x$. This will give us the inverse function $f^{-1}(x)$.

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

A: The relationship between a function and its inverse is that they are two different functions that "undo" each other. 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: 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 it is the only function that "undoes" the original function.

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

A: To graph the inverse of a function, you need to follow these steps:

1. Graph the original function: First, graph the original function $f(x)$.
2. Reflect the graph: Then, reflect the graph of the original function across the line $y = x$.
3. Graph the inverse: Finally, graph the inverse function $f^{-1}(x)$.

Conclusion

In this article, we answered some frequently asked questions about inverse functions. We learned about the difference between a function and its inverse, how to find the inverse of a function, and how to graph the inverse of a function. We also learned about the relationship between a function and its inverse, and how to determine if a function has an inverse.

Final Answer

The final answer is that inverse functions are a fundamental concept in mathematics, and they play a vital role in solving equations, graphing functions, and understanding the behavior of functions.

References

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