To determine if $f(x)$ and $g(x)$ are inverse functions, we need to verify if both compositions $f(g(x)) = x$ and $g(f(x)) = x$ hold true.

Given:

• $f(x) = \sqrt[3]{x + 1}$
• $g(x) = x^3 + 1$

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

We substitute $g(x) = x^3 + 1$ into $f(x)$:

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

Since $\sqrt[3]{x^3 + 2} \neq x$, $f(g(x)) \neq x$.

Conclusion

Since $f(g(x)) \neq x$, $f(x)$ and $g(x)$ are not inverse functions of each other.

The answer is:

• No, they are not inverse functions.

Would you like more details or have any questions about this solution?

Here are 5 related questions to deepen your understanding:

1. How would you find the inverse of a function in general?
2. What are the conditions for two functions to be inverses of each other?
3. How does the cubic root function differ from the square root function?
4. Can inverse functions exist for non-linear functions?
5. What is the role of domain and range in determining inverse functions?

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