TITLE: What is the name for the unreachable part of a function's codomain?
QUESTION [5 upvotes]: A function associates each element from a "domain" set/space with an element from a "codomain" set/space. It does not have to associate all the elements in the codomain. If it does use them all, it is "surjective".
I would like to know, what is the name for the set/space of elements in the codomain that are not used by a non-surjective function?
I am particularly interested in the context of linear mappings. The space I would like the name of is somewhat like the opposite of the nullspace.

REPLY [3 votes]: Given a function $\DeclareMathOperator{\ran}{ran}f:A \to B$, I usually denote $A$ as the domain and $B$ as the codomain. The range of $f$ will always be a subset of the codomain (and will be equal to its codomain if it's surjective). The set you are interested in could just be written as $B\setminus\ran(f)$.
Formally, a function always associates at least one element of its domain to one element in its range. Since the range is defined to be $f(A) =\{f(a) : a\in A\}$ where $A$ is the domain. This formalism is not always distinguished to newer students in math.