# Preimage of a singleton set.

My question is aroused by this article;

"By definition of a function, the image of an element x of the domain is always a single element y of the codomain. Conversely, though, the preimage of a singleton set (a set with exactly one element) may in general contain any number of elements. For example, if f(x) = 7 (the constant function taking value 7), then the preimage of {5} is the empty set but the preimage of {7} is the entire domain."

Here,I can understand that the preimage of the singleton set is the entire domain.But if so,then how does the inverse of a singleton function be a "function" according to the function definition as there exists more than one element map singleton set to preimage.

The preimage, however, is simply a set $f^{-1}(E) = \{x : f(x) \in E \}$, which always makes sense.