Determine whether the function $f(x)=x^2-4x+4$ is one‑to‑one on the set of real numbers. Justify your answer.

Step-by-step solution

1. Express $f(x)$ as $(x-2{)}^2$ to identify it as a perfect square.
2. Compute $f(0)=0^2-0+4=4$ and $f(4)=4^2-16+4=4$; two distinct inputs give the same output.
3. Since $f(0)=f(4)$, the function is not injective.
4. Conclude that $f(x)$ is not one‑to‑one on $\mathbb{R}$.

Concept tested: Injectivity of quadratic function

Published 17 September 2026 · How this solution was prepared and checked