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.