We are tasked with finding the inverse of the function $f(x) = (x + 5)^{\frac{1}{7}}$.

Steps to Find the Inverse:

1. Start with the given function: $y = (x + 5)^{\frac{1}{7}}$ We want to solve for $x$ in terms of $y$ to find the inverse.

2. Raise both sides to the power of 7: $y^7 = x + 5$

3. Solve for $x$: $x = y^7 - 5$

Thus, the inverse function is: $f^{-1}(x) = x^7 - 5$

Correct Answer:

The correct option is $f^{-1}(x) = x^7 - 5$.

Would you like further clarification or a breakdown of any step?

Here are 5 related questions for further exploration:

1. How do you verify that two functions are inverses of each other?
2. What is the domain and range of the original function $f(x)$?
3. How would the inverse change if the exponent were different, say $\frac{1}{3}$?
4. What happens to the graph of a function when you find its inverse?
5. Can all functions be inverted? Why or why not?

Tip: To check if two functions are inverses, compose them $f(f^{-1}(x))$ and $f^{-1}(f(x))$. Both should result in $x$.