# Surjectivity of function over the naturals

I've been doing some study with function types such as injective, surjective and bijective. I came across this question $$g(x) = 2x^2 + 5$$ where $$g : \mathbb N \to \mathbb N$$. I would like some clarification. I believe the function is injective (is monotone increasing) but I am not sure whether it is surjective. Could anyone give me an explanation on whether it is or not?

I will not give away the answer directly but here's how you can proceed. Let $$y \in \Bbb{N}$$. See if there exists an $$x \in \Bbb{N}$$ such that $$2x^2+5=y$$. If such an $$x$$ always exists, then the function is surjective, otherwise it is not.
EDIT: Now that you know it is not surjective, here's a formal proof. For surjectivity, $$\forall y \in \Bbb{N}$$, $$\exists x \in \Bbb{N}$$ such that $$2x^2+5=y$$ $$\implies x^2=\frac{y-5}{2}$$ But $$x^2 \geq 0 \implies \frac{y-5}{2} \geq 0 \implies y-5 \geq 0 \implies y \geq 5$$. Thus for each $$y < 5$$, there is no $$x \in \Bbb{N}$$ which maps to it. Thus, not surjective.
• So, $2x^2 + 5 = 1$, there is no solution and thus is not surjective. Thanks for your explanation! Commented Apr 4, 2020 at 8:55