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.