To solve $f^{-1}(16)$ for the function $f(x) = x^9$, we need to find the inverse function and then evaluate it at 16.

1. Start by setting $f(x) = y$, so $y = x^9$.
2. To find the inverse, solve for $x$ in terms of $y$. Take the 9th root of both sides: $x = \sqrt[9]{y}$ Therefore, the inverse function is: $f^{-1}(y) = \sqrt[9]{y}$
3. Now evaluate $f^{-1}(16)$: $f^{-1}(16) = \sqrt[9]{16}$

This means we need to find the 9th root of 16. Using approximate methods: $\sqrt[9]{16} \approx 1.283$

So, $f^{-1}(16) \approx 1.283$.

Would you like more details on how to approximate the 9th root, or have any other questions?

Here are five related questions to explore further:

1. What is the general method for finding inverse functions?
2. How do we approximate higher-order roots using logarithms?
3. What are other examples of functions with easily found inverses?
4. How would you calculate $f^{-1}(x)$ for different powers of $x$?
5. Can all functions have an inverse? What are the criteria?

Tip: Inverse functions "undo" the original function. For a function to have an inverse, it must be one-to-one, meaning it passes the horizontal line test.