Determine whether the function $f(x)=x^3-3x$ is one‑to‑one on $\mathbb{R}$ and justify your answer.

Step-by-step solution

1. Compute the derivative: $f^{\prime }(x)=3x^2-3=3(x^2-1)$.
2. Find critical points: $f^{\prime }(x)=0$ when $x=pm1$.
3. Evaluate the sign of $f^{\prime }$:
4. • For $x<-1$, $x^2>1$ ⇒ $f^{\prime }>0$ (increasing).
5. • For $-1<x<1$, $x^2<1$ ⇒ $f^{\prime }<0$ (decreasing).
6. • For $x>1$, $x^2>1$ ⇒ $f^{\prime }>0$ (increasing).
7. Since $f$ increases, then decreases, then increases, it is not monotonic.
8. A non‑monotonic cubic with a local max and local min attains some $y$‑values twice.
9. Hence $f$ fails to be one‑to‑one on $\mathbb{R}$.

Concept tested: Injectivity of cubic functions

Published 15 September 2026 · How this solution was prepared and checked