We are given the function $f(x) = \sqrt[5]{x} - 2$, and we are asked to find the inverse function $f^{-1}(x)$.

To find the inverse function, we follow these steps:

1. Start by setting $y = f(x)$, so $y = \sqrt[5]{x} - 2$.

2. Now, solve for $x$ in terms of $y$ by isolating $x$.

   First, add 2 to both sides of the equation: $y + 2 = \sqrt[5]{x}$

3. Now, raise both sides of the equation to the power of 5 to remove the fifth root: $(y + 2)^5 = x$

4. The inverse function is obtained by switching $x$ and $y$: $f^{-1}(x) = (x + 2)^5$

Thus, the correct inverse function is $f^{-1}(x) = (x + 2)^5$.

From the answer choices, the correct option is: $f^{-1}(x) = (x + 2)^5$

Do you need further details or have any questions?

Here are 5 related questions to expand on this concept:

1. How do you verify that two functions are inverses of each other?
2. What is the domain and range of the inverse function $f^{-1}(x)$?
3. How does the behavior of the function $f(x) = \sqrt[5]{x} - 2$ change when you add or subtract constants?
4. What are the general steps to find the inverse of any function?
5. Can every function have an inverse? Why or why not?

Tip: When finding an inverse, make sure to express the original variable completely in terms of the new one and then swap them to get the final form.