Determine whether this function f(n) = n2 − 1 is one-to-one, onto, or both. Explain your answers. The domain of each function is the set of all integers. The codomain of each function is also the set of all integers

Let us determine whether the function $f:\Z\to\Z,\ f(n) = n^2 − 1,$ is one-to-one, onto, or both.

Since $f(-1)=(-1)^2-1=0=1^2-1=f(1),$ we conclude that this function is not one-to-one. Taking into account that for $y=-2$ the equation $f(n)=y,$ that is $n^2-1=-2$ or $n^2=-1,$ has no integer solutions, we conclude that $f^{-1}(-2)=\emptyset,$ and hence the function $f$ is not onto. Therefore, $f$ is not a bijection.