To determine if $g(x)$ is the inverse of $f(x)$, we need to check if the compositions of these functions, $f(g(x))$ and $g(f(x))$, result in the identity function $x$.

Step 1: Composition $f(g(x))$

Given:

• $f(x) = (x + 2)^3$
• $g(x) = \sqrt[3]{x} - 2$

Now, calculate $f(g(x))$: [ f(g(x)) = f\left(\sqrt[3]{x} - 2\right) = \left((\sqrt[3]{x} - 2) + 2\right)^3 = \left(\sqrt[3]{x}\right)^3 = x ] So, $f(g(x)) = x$.

Step 2: Composition $g(f(x))$

Next, calculate $g(f(x))$: $g(f(x)) = g\left((x + 2)^3\right) = \sqrt[3]{(x + 2)^3} - 2 = x + 2 - 2 = x$ Thus, $g(f(x)) = x$.

Since both compositions $f(g(x)) = x$ and $g(f(x)) = x$, we can conclude that $g(x)$ is indeed the inverse of $f(x)$.

So, the answer is "yes".

Would you like further explanation or have any questions?

Here are 5 related questions for deeper understanding:

1. What is the general process to find the inverse of a function?
2. How do you prove that two functions are inverses of each other?
3. What are the domain and range of inverse functions?
4. Can every function have an inverse? Why or why not?
5. What are some real-world applications of inverse functions?

Tip: To confirm two functions are inverses, always check both compositions: $f(g(x)) = x$ and $g(f(x)) = x$.