This might be something absolutely elementary, ridiculous or obvious (or way too complex) but the thing is, I have no clue and have found no information about it. Let's say we have some set resulting from a mapping, and that set contains some element e which has no pre-image in the domain of the function, now we compose this function with one another such that this set defines itself a domain. How can this element e be the pre-image of some element in the new codomain if it has no inverse operator? Isn't it that we do not have enough information about it as to act upon it? I somehow get the idea that in mathematics the behaviour of things is far more important than their presence.
Plus, can somebody recommend me some book or source which will explain elementary stuff like this such that people which really are looking for just elementary stuff will understand it?