MR. ElementOf surprise!
u have written a thing which is mathematically incorrect.
i.e.
if g(f(x)) = something ...that does not imply that dom (g (f(x)) = dom (f(x)).
Consider the following counter example.....
Let f(x) = x^2. then Dom (g(f(x)) = postive real numbers and zero...however dom(f(x)) = All real numbers.