TITLE: What is the preimage of a function
QUESTION [9 upvotes]: Is it the domain of the inverse or the defined domain of the inverse (in the case the function is not invertible)? The concept is very confusing for some reason I cannot grasp.

REPLY [5 votes]: I would not use the terminology preimage of a function at all. If $f:X\to Y$ is a function, and $A\subseteq Y$, the preimage of $A$ under $f$ is $$f^{-1}[A]=\big\{x\in X:f(x)\in A\big\}\;.$$ That is, in my terminology subsets of the codomain of $f$ have preimages under $f$, and these preimages are subsets of the domain of $f$.
You can, if you wish, define a function $$f^{\leftarrow}:\wp(Y)\to\wp(X):A\mapsto f^{-1}[A]$$ that takes each subset of $Y$ to its preimage in $X$ under the function $f$, but I would never call this function the preimage of $f$..