discrete mathematics - Functions images and inverse images

The objective of this question is to find if the function is a bijective function or not and if it is a bijective find its images and inverse images.

$$f:\mathbb{Z^2} \to \mathbb{Z}$$
$$f(n,k) = n^2k$$

We have to find inverses of   $f^{-1}(\{0\})$,  $f^{-1}(\mathbb{N})$  and  $f(\mathbb{Z} \times \{1\})$

But I fail to understand the approach to this problem, I do understand that they need to have unique mappings and co-domains must be matched, but could anyone help me make it analogous to this situation?

questions such $$y = x^2$$ is not bijective since they have multiple images and are not bijective. Their inverse will be a sqaure root with + and - and hence its an invalid case. Could someone please correct my approach?