f^-1(x) = cuberoot (x-9)
f(x) = x^3 + 9

Substitute the definition of​ f(x) and solve: f(f^-1(x))= f() = 
Substitute the definition of​ f^-1(x) and solve: f^-1(f(x)) = f^-1() =

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:
\[
f(f^{-1}(x)) = x
\]

---

### 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:
\[
f^{-1}(f(x)) = x
\]

---

### 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.

---

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.

f^-1(x) = cuberoot(x-9), f(x) = x^3 + 9. Substitute the definition of f(x) and solve: f(f^-1(x)) = f() = . Substitute the definition of f^-1(x) and solve: f^-1(f(x)) = f^-1() =

Explore how to solve and verify inverse functions using f(x) = x^3 + 9 and its inverse f^-1(x) = cuberoot(x-9). Detailed step-by-step solution includes function composition and verifying f(f^-1(x)) = x and f^-1(f(x)) = x. Suitable for students in Grades 9-12.

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