The problem asks to find the inverse $f^{-1}(x)$ for the given function:

$f(x) = \frac{x^{\frac{1}{9}}}{9} + 7$

Steps to find the inverse:

1. Set $f(x)$ to $y$: $y = \frac{x^{\frac{1}{9}}}{9} + 7$ Now, we want to solve for $x$ in terms of $y$.

2. Isolate the term involving $x$: $y - 7 = \frac{x^{\frac{1}{9}}}{9}$

3. Multiply both sides by 9: $9(y - 7) = x^{\frac{1}{9}}$

4. Raise both sides to the 9th power to eliminate the fractional exponent: $\left(9(y - 7)\right)^9 = x$

5. Conclusion: The inverse function $f^{-1}(x)$ is: $f^{-1}(x) = (9(x - 7))^9$

This matches the first option:

$f^{-1}(x) = (9(x - 7))^9$

Would you like more details on the steps or have any questions?

Related Questions:

1. How do you verify that a function and its inverse are correct?
2. What are the properties of inverse functions?
3. How can you find the domain and range of a function and its inverse?
4. What is the graphical relationship between a function and its inverse?
5. How does the process of finding inverses differ when dealing with different types of functions?

Tip: Always check if the function is one-to-one before finding the inverse, since only one-to-one functions have inverses.