Given $f: A\to B$, how do I prove that $f$ is not surjective

Question

How do I prove that the function $$ f(x) = x^2-4x+7 $$ is not surjective under $$f: (2, \infty) \to (2, \infty) $$

Answer 1

Rewrite it as the following:

$$f(x)=(x-2)^2+3$$

Thus the vertex is $(2,3)$ and the function is concave up. Thus, no elements of the interval $[2,3)$ are mapped to, meaning it is not surjective.