Find a formula for the inverse of the following function, if possible.

g(x)=x^1/3−1

The function given is:

\[
g(x) = x^{1/3} - 1
\]

To find the inverse function \( g^{-1}(x) \), follow these steps:

1. **Replace \( g(x) \) with \( y \)**:
   \[
   y = x^{1/3} - 1
   \]

2. **Solve for \( x \) in terms of \( y \)**:
   \[
   y + 1 = x^{1/3}
   \]

   Now cube both sides to eliminate the cube root:
   \[
   (y + 1)^3 = x
   \]

3. **Express \( x \) in terms of \( y \)**:
   \[
   x = (y + 1)^3
   \]

4. **Replace \( y \) with \( x \)** to get the inverse function**:
   \[
   g^{-1}(x) = (x + 1)^3
   \]

Thus, the inverse of the function is:

\[
g^{-1}(x) = (x + 1)^3
\]

### Do you have any questions about this solution, or would you like more details?

Here are five related questions:

1. How can you verify that two functions are inverses of each other?
2. What is the domain and range of the function \( g(x) = x^{1/3} - 1 \)?
3. How does the process of finding the inverse differ for non-linear functions?
4. What happens if a function doesn't have an inverse? How do you check?
5. What are some practical applications of inverse functions?

**Tip:** To check if your inverse is correct, compose the original function with the inverse \( g(g^{-1}(x)) \) and \( g^{-1}(g(x)) \). Both should give you \( x \).

Find a formula for the inverse of the following function, if possible: g(x)=x^1/3−1

Learn how to find the inverse of the function g(x) = x^(1/3) - 1 through a step-by-step solution. This algebraic problem explores inverse functions and cube roots, making it suitable for students in Grades 10-12.

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