Determine if the following functions are inverses.f(x)=x3−1g(x)=3√x+1

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.

Determine if the following functions are inverses. f(x) = x^3 − 1, g(x) = ³√(x + 1)

Learn how to determine whether the functions f(x) = x³ − 1 and g(x) = ³√(x + 1) are inverses by verifying their compositions. This problem involves function inverses, function composition, and algebraic manipulation. Suitable for high school students in Grades 10-12.

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