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

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Introduction

In mathematics, 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. In this article, we will explore the concept of inverse functions and learn how to solve for $f^{-1}(x)$ using a given function $f(x) = 5x$.

What is an Inverse Function?

An inverse function is a function that undoes 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. For example, if we have a function $f(x) = 2x$, its inverse function $f^{-1}(x)$ will take the output of $f(x)$ and return the original input.

How to Find the Inverse of a Function

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

1. Switch the x and y variables: Switch the x and y variables in the original function. This will give us a new function with the x and y variables reversed.
2. Solve for y: Solve the new function for y. This will give us the inverse function.
3. Replace y with $f^{-1}(x)$: Replace y with $f^{-1}(x)$ in the inverse function. This will give us the final inverse function.

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

Now that we have learned how to find the inverse of a function, let's apply this knowledge to the function $f(x) = 5x$. To find the inverse of this function, we need to follow the steps outlined above.

Step 1: Switch the x and y variables

The original function is $f(x) = 5x$. To switch the x and y variables, we need to replace x with y and y with x. This will give us a new function with the x and y variables reversed.

$f(x) = 5x$

Switching the x and y variables, we get:

$x = 5y$

Step 2: Solve for y

Now that we have the new function with the x and y variables reversed, we need to solve for y. To do this, we need to isolate y on one side of the equation.

$x = 5y$

Dividing both sides of the equation by 5, we get:

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

Step 3: Replace y with $f^{-1}(x)$

Now that we have solved for y, we need to replace y with $f^{-1}(x)$. This will give us the final inverse function.

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

Conclusion

In this article, we have learned how to find the inverse of a function using the given function $f(x) = 5x$. We have followed the steps outlined above to find the inverse function, and we have arrived at the final inverse function $f^{-1}(x) = \frac{1}{5}x$. This is the correct answer, and it is option C in the discussion category.

Final Answer

The final answer is:

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

Introduction

In our previous article, we explored the concept of inverse functions and learned how to solve for $f^{-1}(x)$ using a given function $f(x) = 5x$. In this article, we will continue to delve deeper into the world of inverse functions and answer some of the most frequently asked questions about this topic.

Q: What is the purpose of an inverse function?

A: The purpose of an inverse function is to reverse 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: 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 a function passes the horizontal line test, it has an inverse.

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

A: A function and its inverse are two different functions that work together to undo each other's operations. For example, if we have a function $f(x) = 2x$, its inverse function $f^{-1}(x)$ 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, you need to follow these steps:

1. Switch the x and y variables: Switch the x and y variables in the original function. This will give you a new function with the x and y variables reversed.
2. Solve for y: Solve the new function for y. This will give you the inverse function.
3. Replace y with $f^{-1}(x)$: Replace y with $f^{-1}(x)$ in the inverse function. This will give you the final inverse function.

Q: What is the notation for an inverse function?

A: The notation for an inverse function is $f^{-1}(x)$. This notation indicates that the function $f^{-1}(x)$ is the inverse of the function $f(x)$.

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 by the notation $f^{-1}(x)$.

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 work together to undo each other's operations. For example, if we have a function $f(x) = 2x$, its inverse function $f^{-1}(x)$ will take the output of $f(x)$ and return the original input.

Q: Can a function have an inverse if it is not one-to-one?

A: No, a function cannot have an inverse if it is not one-to-one. A function must be one-to-one in order to have an inverse.

Q: What is the significance of the inverse of a function?

A: The inverse of a function is significant because it allows us to undo the operation of the original function. For example, if we have a function $f(x) = 2x$, its inverse function $f^{-1}(x)$ will take the output of $f(x)$ and return the original input.

Conclusion

In this article, we have answered some of the most frequently asked questions about inverse functions. We have learned about the purpose of an inverse function, how to find the inverse of a function, and the notation for an inverse function. We have also learned about the relationship between a function and its inverse, and the significance of the inverse of a function.

Final Answer

The final answer is:

• The purpose of an inverse function is to reverse the operation of another function.
• A function has an inverse if it is one-to-one.
• The notation for an inverse function is $f^{-1}(x)$.
• A function cannot have more than one inverse.
• The relationship between a function and its inverse is that they are two different functions that work together to undo each other's operations.
• A function must be one-to-one in order to have an inverse.
• The inverse of a function is significant because it allows us to undo the operation of the original function.