I was looking more into what it means for a function to be well defined, and I believe I understand it. Suppose we have a function $f:A \rightarrow B$ where $A = \{1,2,3,4\}$ and $B = \{1,2,3\}$ given by $f(1) = 1, f(2) = 2, f(3) = 3$.
However given the construction of f, f is not well defined since everything in A must be mapped onto something in B, and 4 is not mapped to anything?
I would not use the term not well-defined to describe this case of incomplete definition, where some element is not mapped to anything. Not well-defined typically refers to having a complete definition that may be ambiguous in some way, and then one shows that there is no ambiguity to show that it is well-defined. For example, if I map an equivalence class $[x]$ to $f(x)$ under some map $g$, I would need to show that this map is independent of the choice of representative of equivalence class used, that is, if $[x]=[y]$ then $f(x)=f(y)$, so that $g$ is well-defined.