We are given the function $f(x) = 4 + \sqrt[3]{x}$, and we need to find its inverse, $f^{-1}(x)$.

Steps to find the inverse function:

1. Start by replacing $f(x)$ with $y$ for simplicity: $y = 4 + \sqrt[3]{x}$

2. Switch $x$ and $y$ to begin solving for $y$ (this is the inverse step): $x = 4 + \sqrt[3]{y}$

3. Isolate the cube root term by subtracting 4 from both sides: $x - 4 = \sqrt[3]{y}$

4. Cube both sides to eliminate the cube root: $(x - 4)^3 = y$

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

The inverse of $f(x) = 4 + \sqrt[3]{x}$ is $f^{-1}(x) = (x - 4)^3$.

Would you like further details or have any questions about this solution?

Here are 5 related questions:

1. How do you verify if two functions are inverses of each other?
2. Can every function have an inverse?
3. What is the domain and range of the inverse function $f^{-1}(x)$?
4. How does the transformation of shifting and cubing affect the graph of a function?
5. What role does the cube root play in finding the inverse?

Tip: When finding an inverse function, always switch $x$ and $y$, then solve for $y$, which gives the inverse.