Preimage
The preimage of a function \(f:X \rightarrow Y\) is defined to be the set \(f^{-1}(Y)\) of all \(x \in X\) such that \(f(x) \in Y\). That is:
\[ f^{-1}(Y)=\{ x \in X : f(x) \in Y \} \] (sidenote: We remark that the preimage of a function is always a subset of the domain of \(f\). That is: \(f^{-1}(Y) \subseteq X\) )