https://file.aiquickdraw.com/mathbot/file/17254814107079kdjp3xb.jpg

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 \( f^{-1}(f(x)) = x \).

For the function f(x) = ⁵√x + 1, find f⁻¹(x).

Learn how to find the inverse of the function f(x) = ⁵√x + 1 with a detailed step-by-step solution. This problem covers key mathematical concepts such as inverse functions and radical expressions. Suitable for students in Grades 10-12.

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