How to prove function surjective

Question

I must prove that given function is surjective for all natural numbers. How to do that? Please include detailed explanation.

$g:\mathbb{N}\to\mathbb{N}$ $g(x) = x^2+1$

I am trying to solve but I cannot continue: $$y = f(x)$$ $$y = x^2+1$$ $$y -1 = x^2$$ How must I contunue from here?

Answer 1

$g:\mathbb{N}\to\mathbb{N}$ is not surjective.

For example, there is no $x\in\mathbb{N}$ (codomain) such that $g(x)=9$, because $g$ is increasing and $g(2)=5$ and $g(3)=10$, so $9$ is skipped.

Answer 2

Let $n \in \mathbb{N}^{*}$ $$ g\left(x\right)=n \Leftrightarrow x^2=n-1 $$ Then $$ x= \pm \sqrt{n-1} $$

However$\ g(x) \ne 0$ for all $x$ ( except if $x=i$ ) but depends on your domain of $g$.