Give an example of a codomain $S$ so that the function $h: \mathbb{N} \rightarrow S$ is bijective.
\begin{align*} h(n) &= \begin{cases} n & n \textrm{ is even.}\\ -2n & n \textrm{ is odd.} \end{cases} \end{align*}
I think that the codomain would be all even integers except 0 and {-4,-8,-12...}. Is this correct? $S=\{2k:k\in \mathbb{Z} \} \setminus \{-4k:k\in \mathbb{N} \} \setminus \{0\}$.
Hint: list the images $h(n)$ for some values of $n$, like $h(0)=0$, $h(1)=-2$, $h(2)=2$, $h(3)=-6$, $h(4)=4$, etc. Do you notice a pattern?
(In fact, looking at your earlier question might be another hint.)