1. Is the function 𝑓: ℤ → ℤ 𝑓(𝑥) = 𝑥 2 + 3 injective, surjective or bijective? Prove your assertions

Consider the function $f:\Z\to\Z,\ f(x)=x^2+3.$ Since for $x_1=-1$ and $x_2=1\ne x_1$ we have that $f(x_1)=(-1)^2+3=4=1^2+3=f(x_2),$ we conclude that the function $f$ is not injective. Taking into account that for $y=0\in\Z$ the equation $x^2+3=0$ has no integer roots, we conclude that $f^{-1}(0)=\emptyset,$ and hence the function $f$ is not surjective. Therefore, the function $f$ is not bijective.