Range of a parabola

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Consider $f:\mathbb{R}\to\mathbb{R}$, $f(x)=x^{2}-4x+7$. Find the range and decide whether $f$ is surjective.

Solution

Complete the square: $f(x)=x^{2}-4x+7=(x-2)^{2}+3.$ Since $(x-2)^{2}\ge 0$, we have $f(x)\ge 3$, with minimum $3$ attained at $x=2$. As $|x|$ grows the function grows without bound, so the range is $[3,+\infty)$. Not every real number is a value (for instance $0$ is not), so $f$ is not surjective on $\mathbb{R}$. $\boxed{\,\operatorname{Im}f=[3,+\infty),\quad y_{min}=3\ \text{at}\ x=2\,}$