To find the inverse function $f^{-1}(x)$ for the given function $f(x) = \left( \frac{x - 7}{7} \right)^{\frac{1}{5}} + 9$, we will follow these steps:

1. Rewrite the function as an equation: $y = \left( \frac{x - 7}{7} \right)^{\frac{1}{5}} + 9$

2. Swap $x$ and $y$ to find the inverse: $x = \left( \frac{y - 7}{7} \right)^{\frac{1}{5}} + 9$

3. Isolate $y$: $x - 9 = \left( \frac{y - 7}{7} \right)^{\frac{1}{5}}$

4. Raise both sides to the power of 5 to eliminate the fractional exponent: $(x - 9)^5 = \frac{y - 7}{7}$

5. Multiply both sides by 7: $7(x - 9)^5 = y - 7$

6. Add 7 to both sides to solve for $y$: $y = 7(x - 9)^5 + 7$

So, the inverse function is: $f^{-1}(x) = 7(x - 9)^5 + 7$

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

1. How do you find the inverse of a function in general?
2. Can you show how to find the derivative of the inverse function?
3. What are the properties of inverse functions?
4. How can you verify that two functions are inverses of each other?
5. Can you explain the domain and range of the inverse function?

Tip: When finding the inverse of a function, always remember to swap $x$ and $y$ and then solve for $y$.