To find the inverse function $f^{-1}(x)$ of the function $f(x) = 9 - x$, we follow these steps:

1. Start with the equation: $y = 9 - x$

2. Switch $x$ and $y$: $x = 9 - y$

3. Solve for $y$: $y = 9 - x$

   Therefore, the inverse function is: $f^{-1}(x) = 9 - x$

So, $f^{-1}(x) = 9 - x$.

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. What is the graphical interpretation of the inverse function?
2. How do you verify that two functions are inverses of each other?
3. How does the inverse function relate to the identity function?
4. What are the conditions under which a function has an inverse?
5. How would the inverse function change if $f(x)$ had a different form?

Tip: When finding the inverse of a function, always make sure that the original function is one-to-one, meaning it passes both the horizontal and vertical line tests. This ensures that the inverse is also a function.