Let's solve the problem step by step. We are asked to find the inverse function $f^{-1}(x)$ for the given function $f(x) = \sqrt[3]{x} + 4$.

Steps to Find the Inverse Function:

1. Express the function: Start with the equation $y = \sqrt[3]{x} + 4$.

2. Swap $x$ and $y$: To find the inverse, we switch $x$ and $y$: $x = \sqrt[3]{y} + 4$

3. Solve for $y$:

   • First, subtract 4 from both sides to isolate the cube root: $x - 4 = \sqrt[3]{y}$
   • Then, cube both sides to get rid of the cube root: $(x - 4)^3 = y$
   • So, the inverse function is: $f^{-1}(x) = (x - 4)^3$

Answer:

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

Would you like more details on this or have any further questions?

Here are five related questions to consider:

1. How would you find the inverse of a quadratic function?
2. What is the importance of finding the domain and range of inverse functions?
3. Can a function always have an inverse? Why or why not?
4. What is the graphical relationship between a function and its inverse?
5. How do you verify that two functions are inverses of each other?

Tip: When finding the inverse of a function, remember that you essentially reverse the operations performed on the variable.