Solution

Consider the function $f: \mathbb{Z} \rightarrow 2\mathbb{Z}$ defined by $f(x) = 2x^2 - 4$. Is $f(x)$ injective? Is it surjective? Justify your claims.

$f(x)$ is not injective as we can find two different values in $\mathbb{Z}$ (the domain) that produce the same output (e.g., $f(-1) = -2 = f(1)$).

$f(x)$ is not surjective as we can find elements of $2\mathbb{Z}$ (which is called the "co-domain") that fail to be output values of $f(x)$. For example, there are no integers $x$ with $f(x) = 0$, $f(x)=2$, $f(x)=6$, etc...