To solve for $f^{-1}(x)$, we need to find the inverse of the given function $f(x) = \frac{x^{1/3}}{5}$.

Step-by-Step Process:

1. Start with the function: $y = \frac{x^{1/3}}{5}$

2. Swap $x$ and $y$ to find the inverse: $x = \frac{y^{1/3}}{5}$

3. Isolate $y$: Multiply both sides by 5: $5x = y^{1/3}$

   Raise both sides to the power of 3 to eliminate the cube root: $y = (5x)^3$

4. The inverse function is: $f^{-1}(x) = (5x)^3$

Final Answer:

The correct option is: $f^{-1}(x) = (5x)^3$

Would you like a detailed explanation of why this process works or have any questions about function inverses?

Related Questions:

1. What is the general procedure for finding the inverse of a function?
2. How do you determine if a function is invertible?
3. What does $f(f^{-1}(x)) = x$ mean, and why does it hold?
4. Can all functions have inverses? Why or why not?
5. How would the process differ if the function involved a square root instead of a cube root?

Tip:

Always verify an inverse function by substituting it into the original function to ensure $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.