Let $R$ be a binary relation. If $A\subseteq\dom R$, the image of $A$ under $R$ is
\[R[A] = \{y:(\exists x\in A:x\,R\,y)\}.\]This is a set:
\[R[A] \subseteq \ran R.\]The image of $R$ is just a synonym of its range:
\[\im R = \ran R.\]If $B\subseteq\ran R$, the inverse image / preimage of $B$ under $R$ is $R^{-1}[B]$.
For any binary relation $R$,
\[R[\varnothing] = \varnothing.\]