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The function given is \( f(x) = x^3 + 2 \), and you're tasked with finding the inverse function \( f^{-1}(x) \).

### To find the inverse function \( f^{-1}(x) \), follow these steps:
1. **Step 1: Replace \( f(x) \) with \( y \):**
   \[
   y = x^3 + 2
   \]
2. **Step 2: Solve for \( x \) in terms of \( y \):**
   \[
   y - 2 = x^3
   \]
   \[
   x = \sqrt[3]{y - 2}
   \]
3. **Step 3: Swap \( x \) and \( y \) to express the inverse:**
   \[
   f^{-1}(x) = \sqrt[3]{x - 2}
   \]

Thus, the inverse function is:
\[
f^{-1}(x) = \sqrt[3]{x - 2}
\]

Would you like further clarification or details on checking the answer?

Here are 5 related questions to explore this topic further:
1. How do you verify that \( f(f^{-1}(x)) = x \) for this inverse function?
2. What are the steps to find the domain of the inverse function \( f^{-1}(x) \)?
3. How can you find the range of the original function \( f(x) \)?
4. How does the graph of \( f(x) \) relate to the graph of its inverse function \( f^{-1}(x) \)?
5. What is the significance of the line \( y = x \) when graphing a function and its inverse?

**Tip:** When finding the inverse of a function, always check that the function is one-to-one, ensuring it has an inverse.

The function f(x) = x^3 + 2 is one-to-one. (a) Find the inverse of f and check the answer.

This tutorial explains how to find the inverse of the cubic function f(x) = x^3 + 2. We walk through solving for the inverse and verifying the result. Suitable for high school students studying advanced algebra topics.

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