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Given the function \( f(x) = 5x \), we are asked to find the inverse function \( f^{-1}(x) \).

To find the inverse of a function, we swap the roles of \( x \) and \( y \) and then solve for \( y \). Here are the steps:

1. Start with the original function \( f(x) = 5x \). Set \( y = 5x \).
   \[
   y = 5x
   \]
2. Swap \( y \) and \( x \) (because \( f^{-1}(x) \) undoes what \( f(x) \) does):
   \[
   x = 5y
   \]
3. Solve for \( y \) (which will be the inverse function):
   \[
   y = \frac{x}{5} = \frac{1}{5}x
   \]

Thus, the inverse function is \( f^{-1}(x) = \frac{1}{5}x \).

The correct answer is:
\[
f^{-1}(x) = \frac{1}{5}x
\]

Would you like more details on how inverses of functions are derived or have any other questions?

Here are some related questions:
1. What is the general method for finding the inverse of a linear function?
2. How do you determine if a function has an inverse?
3. Can a function be its own inverse? Provide an example.
4. What is the relationship between a function and its inverse on a graph?
5. What are the properties of inverse functions in terms of composition?

**Tip:** To check if two functions are inverses, their compositions should equal the identity function: \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

Explore how to find the inverse of the linear function f(x) = 5x with clear explanations and mathematical steps. This problem involves understanding the concepts of algebra, functions, and specifically, inverse functions. Suitable for high school students studying algebra.

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