Is it legal to say $f(E)=\emptyset$ if set $E$ not in the function $f$ domain In many inverse functions, I have seen $f^{-1}(E)=\emptyset$, where the set $E$ is not in the function $f$ range.
So, is it also right to say $f(E)=\emptyset$ if set $E$ not in the function domain.

The empty set $\phi$  is a well defined set. 
On the other hand if $E$ is not a subset of the domain of $f$ the expression $f(E)$ is not defined. 
Therefore in my opinion the answer is no $f(E)$ is not the empty set because it is not a set. 
