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

Substitute the definition of​ f(x): f(f^-1(x))= f() = x^3+9
Substitute the definition of​ f^-1(x): f^-1() = x

We are given the inverse function \( f^{-1}(x) = \sqrt[3]{x} - 9 \), and we are also asked to find the function \( f(x) \), such that:

- \( f(f^{-1}(x)) = x \)
- \( f^{-1}(f(x)) = x \)

### Step 1: Define the function \( f(x) \)
We are given \( f^{-1}(x) = \sqrt[3]{x} - 9 \). To find \( f(x) \), we need to reverse the operations done by \( f^{-1}(x) \). 

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

To isolate \( x \), add 9 to both sides:
\[
f^{-1}(x) + 9 = \sqrt[3]{x}
\]

Now, cube both sides to undo the cube root:
\[
(f^{-1}(x) + 9)^3 = x
\]

This implies that \( f(x) = (x + 9)^3 \).

### Step 2: Verify by substitution

#### Verify \( f(f^{-1}(x)) = x \)
We substitute \( f^{-1}(x) \) into \( f(x) \):
\[
f(f^{-1}(x)) = \left( \left( \sqrt[3]{x} - 9 \right) + 9 \right)^3
\]
Simplifying:
\[
f(f^{-1}(x)) = (\sqrt[3]{x})^3 = x
\]

Thus, \( f(f^{-1}(x)) = x \), as required.

#### Verify \( f^{-1}(f(x)) = x \)
Now, we substitute \( f(x) \) into \( f^{-1}(x) \):
\[
f^{-1}(f(x)) = \sqrt[3]{(x + 9)^3} - 9
\]
Simplifying:
\[
f^{-1}(f(x)) = x + 9 - 9 = x
\]

Thus, \( f^{-1}(f(x)) = x \), as required.

### Final Answer:
The function \( f(x) \) is \( f(x) = (x + 9)^3 \), and the inverse is \( f^{-1}(x) = \sqrt[3]{x} - 9 \).

Would you like more details on any step?

### Related Questions:
1. How do you compute the inverse of a general cubic function?
2. What are the general properties of inverse functions?
3. How can you verify if two functions are inverses of each other?
4. What transformations are involved in the function \( f(x) = (x + 9)^3 \)?
5. How do cube roots work in solving equations?

**Tip:** When working with inverse functions, always verify by checking \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

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

This detailed solution explores how to find the inverse of the cubic function f(x) = (x + 9)^3, verify f(f^-1(x)) = x, and understand inverse functions through algebraic manipulation. Suitable for Grades 10-12 students.

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