Often we are interested in the element(s) whose image is a particular element \(y\) of in the codomain. The notation above works: \(f\inv(\{y\})\) is the set of all elements in the domain that \(f\) sends to \(y\text{.}\) It makes sense to think of this as a set: there might not be anything sent to \(y\) (if \(y\) is not in the range), in which case \(f\inv(\{y\}) = \emptyset\text{.}\) Or \(f\) might send multiple elements to \(y\) (if \(f\) is not injective). As a notational convenience, we usually drop the set braces around the \(y\) and write \(f\inv(y)\) instead for this set.
in-context