1. **State the problem:** Find the inverse function of $g(x) = x^3$. 2. **Recall the definition of inverse function:** The inverse function $g^{-1}(x)$ satisfies $g(g^{-1}(x)) = x$ and $g^{-1}(g(x)) = x$. 3. **Set $y = g(x)$:** $$y = x^3$$ 4. **To find the inverse, solve for $x$ in terms of $y$:** $$x = \sqrt[3]{y}$$ 5. **Replace $y$ with $x$ to write the inverse function:** $$g^{-1}(x) = \sqrt[3]{x}$$ 6. **Explanation:** The cube root function reverses the effect of cubing a number, so it is the inverse. **Final answer:** $$g^{-1}(x) = \sqrt[3]{x}$$