Can a function that is not by definition a function have an "inverse"?

Question

The function f: N -> N, where f(n) = n - 2 I know is not a function (since 0 maps to -2 and -2 is not an element of the Naturals).

Is it possible for the inverse (let's say g) of f to be g : N -> N, g(n) = n + 2 (g is by definition a function) or is there no inverse because f is not by definition a function?

Thanks

Answer 1

If you define $f$: $\mathbb N\to \mathbb N$, it is not bijective, so we can't speak about its inverse.

but if you define $f$: $\mathbb N\setminus{\{0,1\}} \to \mathbb N$, then it is a bijection and its inverse function is given by

$g:\mathbb N\to\mathbb N\setminus{\{0,1\}}$

$\;\;\;n\mapsto n+2$.