definition of a set function?

Question

Assuming $A$ is a set, then $F: A\rightarrow \mathbb{R}$. We can define $F(\varnothing) = 0$. But what is $F(x)$ for $x\notin A$? Is it 0?

Answer 1

For any function $F:A \rightarrow B$, the set $A$ is the domain of $F$.

By definition, $F$ assigns elements in $B$ to elements in $A$. In other words, $F(x)$ is not defined for any $x$ which is not in $A$. Or more informally, it has no meaning: you don't use $F$ to evaluate things which are not in its domain.