Let $f:\mathbb{R}\to \mathbb{R}$ be defined by $f(x)=\{\begin{array}{ll}{x}^3,& x\ge 0\\ -x^3,& x<0\end{array}.\text{Find the inverse function }{f}^{-1}\text{ and state its domain and range}.$

Step-by-step solution

1. Observe that for $x\ge 0$, $f(x)=x^3$ is strictly increasing; for $x<0$, $f(x)=-x^3$ is strictly decreasing.
2. Solve $y=x^3$ for $x\ge 0$ to get $x=\sqrt[3]{y}$.
3. Solve $y=-x^3$ for $x<0$ to get $x=-\sqrt[3]{-y}$.
4. Combine the two cases to define $f^{-1}$ piecewise.
5. Since $x$ can be any real number, the domain and range of $f^{-1}$ are both $\mathbb{R}$.

Concept tested: Inverse of piecewise function

Published 18 September 2026 · How this solution was prepared and checked