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The inverse function \( f^{-1}(x) \) for the given function \( f(x) = \sqrt[7]{\frac{x + 1}{9}} - 3 \) is:

\[
f^{-1}(x) = 9 \left( x + 3 \right)^7 - 1
\]

This matches the last option from the provided answer choices:

\[
f^{-1}(x) = 9(x + 3)^7 - 1
\]

Would you like more details on how the inverse was derived, or do you have any questions?

Here are 5 related questions:
1. How do you verify if a function and its inverse are correct?
2. What is the domain of the function \( f(x) \)?
3. How do transformations (such as shifts and stretches) affect finding inverses?
4. What is the graphical interpretation of a function and its inverse?
5. How does one algebraically check that two functions are inverses?

**Tip**: Always check that the composition of a function and its inverse yields the identity function \( f(f^{-1}(x)) = f^{-1}(f(x)) = x \).

For the function f(x) = 7√(x+1)/9 - 3, find f^(-1)(x).

Learn how to find the inverse function for f(x) = 7√(x+1)/9 - 3. This problem involves understanding inverse functions, manipulating radicals, and algebraic transformations. Suitable for students in Grades 11-12.

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