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 $f^{-1}(f(x)) = x$.