To determine if two functions are inverses, we need to check if their compositions $f(g(x)) = x$ and $g(f(x)) = x$ hold true.

Given functions:

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

Let's check the compositions:

1. Check $f(g(x))$:

We substitute $g(x) = \sqrt[3]{x + 1}$ into $f(x) = x^3 - 1$:

[ f(g(x)) = f\left( \sqrt[3]{x + 1} \right) = \left( \sqrt[3]{x + 1} \right)^3 - 1 ] Since $\left( \sqrt[3]{x + 1} \right)^3 = x + 1$, we have: $f(g(x)) = x + 1 - 1 = x$

2. Check $g(f(x))$:

Now, substitute $f(x) = x^3 - 1$ into $g(x) = \sqrt[3]{x + 1}$:

$g(f(x)) = g\left( x^3 - 1 \right) = \sqrt[3]{(x^3 - 1) + 1}$ This simplifies to: $g(f(x)) = \sqrt[3]{x^3} = x$

Conclusion:

Since both compositions, $f(g(x)) = x$ and $g(f(x)) = x$, hold true, the functions $f(x) = x^3 - 1$ and $g(x) = \sqrt[3]{x + 1}$ are inverses of each other.

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Would you like more details on how the compositions work, or is there anything specific you'd like to explore further?

Here are some related questions to extend your understanding:

1. How do you determine if two functions are inverses algebraically without directly calculating compositions?
2. What happens if the functions involve rational exponents instead of integer exponents?
3. How do inverse functions relate to symmetry across the line $y = x$?
4. Can a function be its own inverse? Provide an example.
5. How does the derivative of an inverse function relate to the original function's derivative?

Tip: When checking if two functions are inverses, always verify both compositions $f(g(x)) = x$ and $g(f(x)) = x$ before concluding.