Determine whether the function $f(x)=x^2-4x+7$ is one‑to‑one on $mathbbR$ and justify your answer.

Step-by-step solution

1. A function is one‑to‑one if its graph is strictly increasing or strictly decreasing over its domain, equivalently its derivative has a constant sign.
2. Compute the derivative: $f^{\prime }(x)=2x-4$ which is zero at $x=2$ and negative for $x<2$ and positive for $x>2$.
3. Thus $f^{\prime }(x)$ changes sign, so $f$ is decreasing on $(-infty,2]$ and increasing on $[2,infty)$.
4. A change in monotonicity means $f$ takes the same value at two different points (e.g. $f(0)=f(4)=7$).
5. Therefore $f$ is not one‑to‑one on $mathbbR$.

Concept tested: One‑to‑one test via derivative

Published 14 September 2026 · How this solution was prepared and checked