I came across this question on StackExchange and it left me with a question.
Consider a function $f(x)$. Consider another function $g(t)$, where $t\in Z$. How would you represent the domain of $f(x)$ if $x\in N - {g(t)}$ as a function?
For example, lets say the range of $g(t)$ is $\{1, 3, 5, 7 ...\}$. Hence the domain of $f(x)$ will be $\{2, 4, 6, 8...\}$ which can be represented by $\phi(n) = 2n,\ n\in N$.
I considered composite functions, but got stuck.
I assume from the context that it's possible to think of $f$ as having domain all of $\mathbb{Z}$ but for some reason you want to consider it only for the odd integers because $g$ maps to even integers from somewhere.
In works, you want the restriction of $f$ to the complement of the range of $g$. You could write that as $$ f: \mathbb{Z} \setminus \text{range}(g) \to \text{whatever}. $$
If this is something you actually need (rather than just a curiousity) I think you should use the words, or, if the integer example is the actual use, just say the domain of $f$ is the odd integers.
