To find the inverse function $f^{-1}(x)$ for the given function $f(x) = \sqrt[3]{\frac{x - 5}{9}}$, we will follow these steps:

1. Rewrite the function as an equation: $y = \sqrt[3]{\frac{x - 5}{9}}$

2. Swap $x$ and $y$ to find the inverse: $x = \sqrt[3]{\frac{y - 5}{9}}$

3. Cube both sides to eliminate the cube root: $x^3 = \frac{y - 5}{9}$

4. Multiply both sides by 9: $9x^3 = y - 5$

5. Add 5 to both sides to solve for $y$: $y = 9x^3 + 5$

So, the inverse function is: $f^{-1}(x) = 9x^3 + 5$

Do you want details or have any questions? Here are five more questions you can ask:

1. How do you find the domain and range of an inverse function?
2. Can you show an example of verifying an inverse function?
3. What are some real-life applications of inverse functions?
4. How do you differentiate inverse trigonometric functions?
5. Can you explain the horizontal line test for determining if a function has an inverse?

Tip: When dealing with roots and exponents in functions, carefully apply the necessary algebraic operations to isolate the variable when finding the inverse.