To find the inverse of the function $f(x) = 10 + x^3$, follow these steps:

1. Start by rewriting the function: $y = 10 + x^3$ where $f(x) = y$.

2. Swap $x$ and $y$ to find the inverse: $x = 10 + y^3$

3. Solve for $y$: $x - 10 = y^3$ $y = \sqrt[3]{x - 10}$

So, the inverse function is: $f^{-1}(x) = \sqrt[3]{x - 10}$

Would you like more details on how this process works?

Here are some related questions:

1. How do you verify if two functions are inverses of each other?
2. What are the domain and range of the inverse function $f^{-1}(x)$?
3. How do the transformations of a function affect its inverse?
4. How do you graph a function and its inverse?
5. How does the cubic root function behave as compared to the cubic function?

Tip: When solving for an inverse function, always swap $x$ and $y$, and then isolate $y$.