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We are asked to find the inverse of the given function:

\[
f(x) = (x + 2)^3 - 8
\]

### Steps to Find the Inverse:

1. **Replace \( f(x) \) with \( y \)**:
   \[
   y = (x + 2)^3 - 8
   \]
   
2. **Swap \( x \) and \( y \)** (this step begins the process of finding the inverse):
   \[
   x = (y + 2)^3 - 8
   \]

3. **Solve for \( y \)**:
   - Add 8 to both sides:
     \[
     x + 8 = (y + 2)^3
     \]
   - Take the cube root of both sides:
     \[
     \sqrt[3]{x + 8} = y + 2
     \]
   - Subtract 2 from both sides:
     \[
     y = \sqrt[3]{x + 8} - 2
     \]

Thus, the inverse function is:

\[
f^{-1}(x) = \sqrt[3]{x + 8} - 2
\]

Would you like more details on how to verify this result or further clarification on the steps?

### Related Questions:
1. What does it mean for a function to be "one-to-one"?
2. How can you verify that two functions are inverses of each other?
3. What is the domain of the inverse function \( f^{-1}(x) \)?
4. How do transformations affect the graph of a cubic function?
5. Can a function have more than one inverse?

### Tip:
Always check if the function is one-to-one by verifying that it passes the horizontal line test; this ensures the function has an inverse.

The function f is one-to-one. Find its inverse: f(x) = (x + 2)^3 - 8

Learn how to find the inverse of the cubic function f(x) = (x + 2)^3 - 8. This step-by-step guide includes solving for y, switching variables, and using the cube root. Suitable for students in Grades 10-12.

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