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To find the inverse of \( n(x) = \sqrt[3]{x + 8} \), we need to follow a systematic process:

### Step 1: Replace \( n(x) \) with \( y \)
Let:
\[
y = \sqrt[3]{x + 8}
\]

### Step 2: Solve for \( x \)
Cube both sides to eliminate the cube root:
\[
y^3 = x + 8
\]
Now, solve for \( x \):
\[
x = y^3 - 8
\]

### Step 3: Replace \( y \) with \( x \)
To express the inverse function, switch \( y \) and \( x \):
\[
n^{-1}(x) = x^3 - 8
\]

### Step 4: Match with the options
The correct inverse function is:
\[
n^{-1}(x) = x^3 - 8
\]
This matches **option A**.

Would you like more details or have any further questions?

Here are some related questions you might find useful:
1. What are the steps to verify that a function and its inverse are correct?
2. How do you find the inverse of a square root function?
3. What is the geometric interpretation of a function and its inverse?
4. How do you determine if a function has an inverse?
5. Can every cubic function have an inverse?

**Tip:** To verify an inverse function, check if \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

Learn how to find the inverse of the cubic root function n(x) = ³√(x + 8) with a step-by-step solution. This problem involves inverse functions, cube roots, and algebraic manipulation. Suitable for students in Grades 9-12.

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