To find the inverse of the function $f(x) = \sqrt[5]{x} + 1$, we can follow these steps:

1. Rewrite the function: Start by setting $y = f(x)$. $y = \sqrt[5]{x} + 1$

2. Isolate the radical: Subtract 1 from both sides to isolate the fifth root. $y - 1 = \sqrt[5]{x}$

3. Remove the radical: Raise both sides to the power of 5 to eliminate the root. $(y - 1)^5 = x$

4. Solve for $x$: Since $y = f(x)$, the inverse function $f^{-1}(x)$ will switch the roles of $x$ and $y$. $f^{-1}(x) = (x - 1)^5$

So, the correct answer is:

$f^{-1}(x) = (x - 1)^5$

This matches the first option provided in the image.

Would you like any further details or explanations on how this works?

Relative Questions:

1. How do you generally find the inverse of a function?
2. What are the conditions under which a function has an inverse?
3. How can you verify that two functions are indeed inverses of each other?
4. What is the significance of one-to-one functions in finding inverses?
5. How does the domain and range of a function relate to its inverse?

Tip:

When finding the inverse of a function, it's often helpful to check the original function and its inverse by composing them, i.e., check that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.