To find the inverse of the function $f(x) = x^3$, we need to solve for $x$ in terms of $y$, where $y = f(x)$.

Steps:

1. Start with the equation: $y = x^3$
2. To find the inverse, solve for $x$ in terms of $y$. Take the cube root of both sides: $x = \sqrt[3]{y}$
3. Therefore, the inverse function is: $f^{-1}(x) = \sqrt[3]{x}$

Thus, the inverse of $f(x) = x^3$ is $f^{-1}(x) = \sqrt[3]{x}$.

Would you like more details or have any questions about this?

Here are 5 related questions to expand on this concept:

1. What is the domain and range of $f(x) = x^3$ and its inverse?
2. How do you verify that two functions are inverses of each other?
3. What is the geometric relationship between a function and its inverse?
4. How does the inverse function behave with respect to odd and even functions?
5. How does the derivative of $f(x) = x^3$ compare to its inverse function?

Tip: To verify if a function and its inverse are correct, check if $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.