To verify that a function and its inverse are correct, we need to check the conditions:

1. $f(f^{-1}(x)) = x$
2. $f^{-1}(f(x)) = x$

Given the function $f(x) = \sqrt[3]{x - 9}$, let's find its inverse $f^{-1}(x)$ first.

Finding the Inverse Function $f^{-1}(x)$:

1. Start with the original function: $y = \sqrt[3]{x - 9}$
2. Swap $x$ and $y$ to find the inverse: $x = \sqrt[3]{y - 9}$
3. Cube both sides to get rid of the cube root: $x^3 = y - 9$
4. Solve for $y$: $y = x^3 + 9$

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

Verifying $f(f^{-1}(x)) = x$:

Now, substitute $f^{-1}(x)$ into $f(x)$:

$f(f^{-1}(x)) = f(x^3 + 9)$

Applying the function $f$:

[ f(x^3 + 9) = \sqrt[3]{(x^3 + 9) - 9} = \sqrt[3]{x^3} = x ]

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

Verifying $f^{-1}(f(x)) = x$:

Now, substitute $f(x)$ into $f^{-1}(x)$:

$f^{-1}(f(x)) = f^{-1}(\sqrt[3]{x - 9})$

Applying the inverse function $f^{-1}$:

[ f^{-1}(\sqrt[3]{x - 9}) = (\sqrt[3]{x - 9})^3 + 9 = x - 9 + 9 = x ]

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

Both conditions are satisfied, so the function and its inverse are correct.

Would you like more details or have any questions? Here are some related questions for further exploration:

1. What is the domain and range of the function $f(x) = \sqrt[3]{x - 9}$?
2. How do you graph the function and its inverse?
3. How do cube roots affect the shape and properties of the function?
4. What are the properties of inverse functions in general?
5. How do you verify inverses of more complex functions?

Tip: When finding inverses, always swap $x$ and $y$, and solve for the new $y$. This technique works universally for basic functions.