Question

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

Answer 1

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.