Let's solve this step by step.

Problem 1: Solve $f(f^{-1}(x))$

We are given that:

• $f(x) = x^3 + 9$
• $f^{-1}(x) = \sqrt[3]{x - 9}$ (the cube root of $x - 9$)

We need to substitute $f^{-1}(x)$ into $f(x)$ to get $f(f^{-1}(x))$.

Step-by-step substitution:

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

Now substitute into $f(x) = x^3 + 9$: $f(f^{-1}(x)) = \left( \sqrt[3]{x - 9} \right)^3 + 9$

Since the cube and cube root cancel each other: $f(f^{-1}(x)) = x - 9 + 9 = x$

Thus, the result is: $f(f^{-1}(x)) = x$

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Problem 2: Solve $f^{-1}(f(x))$

Now we substitute $f(x)$ into $f^{-1}(x)$.

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

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

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

Since the cube root and cube cancel each other: $f^{-1}(f(x)) = x$

Thus, the result is: $f^{-1}(f(x)) = x$

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Summary of results:

• $f(f^{-1}(x)) = x$
• $f^{-1}(f(x)) = x$

These results confirm that $f(x)$ and $f^{-1}(x)$ are indeed inverse functions.

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Do you want further details or have any questions?

Related Questions:

1. What is the general method to find the inverse of a function?
2. How do you verify if two functions are inverses?
3. What conditions must a function satisfy to have an inverse?
4. What happens if you apply a function and its inverse in reverse order?
5. Can you explain the concept of one-to-one and onto functions?

Tip:

When verifying if two functions are inverses, always check both $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ to ensure completeness.