Can a model of a sentence in predicate logic contain a function that can map to a member not contained in the domain of that model?
Example. Is this interpretation correct : domain= $\{1,2\}$ ; sentence=$'(∃x)(∃y)(P(f(xy)))'$ ; $f(xy) =$ the the product of $x$ and $y$ ; $P(x)=x$ is positive. The sentence seems true, but on an assignment $(x=2$ and $y=2)$, the function maps to $4$, which is not contained in the domain.
EDIT : Second question : How would we qualify this model? True, false, neither of those?
The simple answer is No; interpretations of function symbols must be functions that map objects from the domain to the domain, so you can't map to anything outside the domain.
Also, your sentence should be $\exists x \exists y P(f(x,y))$