I am pretty sure this function checks out as bijective.
$f : Z * N \mapsto Q, f(x,y) = \frac{x}{y+1}$
Where Z is the set of all real numbers and N is the set of all non-negative numbers.
The thing that confuses me is the cases where the fraction will simplify, especially to one. If it simplifies on both sides does that make it injective still?
Main question: IF it is bijective, how would I write $f^{-1}$?
It is not bijective since we have
$f(1,1)=f(2,3)=\frac{1}{2}$