Is the function $f: R \to R$ defined by $y=x^{2}-2x-2$ a surjection?

Question

The problem is to find if the following function is a surjection.

$f: R\to R$ defined by $y=x^{2}-2x-2$

I know that it is not a surjection by looking at a graph of the function but I am new to formal proofs. How would I show that it is not onto (surjection) formally?

Answer 1

For each $x\in\mathbb R$\begin{align}f(x)&=x^2-2x-2\\&=x^2-2x+1-3\\&=(x-1)^2-3\\&\geqslant-3.\end{align}So, the range of $f$ is a subset of $[-3,\infty)$ (actually, it's equal to it) and therefore $f$ is not surjective.

Answer 2

f surjective means: $\forall y \in \mathbb{R},\exists x \in \mathbb{R}:f(x)=y$ meaning that $f(x)=y$ must have solution for any real number y. Since the range of f is limited to a subset of $\mathbb{R}$, there exists a value of y outside the range of f and still a real number for which the equation $f(x)=y$ has no solution. Therefore f is not surjective