To find the inverse of the function \( f(x) = x^3 \), we need to solve for \( x \) in terms of \( y \), where \( y = f(x) \).

### Steps:
1. Start with the equation:
   \[
   y = x^3
   \]
2. To find the inverse, solve for \( x \) in terms of \( y \). Take the cube root of both sides:
   \[
   x = \sqrt[3]{y}
   \]
3. Therefore, the inverse function is:
   \[
   f^{-1}(x) = \sqrt[3]{x}
   \]

Thus, the inverse of \( f(x) = x^3 \) is \( f^{-1}(x) = \sqrt[3]{x} \).

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

Here are 5 related questions to expand on this concept:
1. What is the domain and range of \( f(x) = x^3 \) and its inverse?
2. How do you verify that two functions are inverses of each other?
3. What is the geometric relationship between a function and its inverse?
4. How does the inverse function behave with respect to odd and even functions?
5. How does the derivative of \( f(x) = x^3 \) compare to its inverse function?

**Tip:** To verify if a function and its inverse are correct, check if \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

Learn how to find the inverse of the cubic function f(x) = x^3 through a detailed step-by-step solution. This problem covers key mathematical concepts such as inverse functions and the inverse function theorem, making it suitable for students in Grades 9-12.

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