How to describe an instance where every $y$ in a codomaine $Y$ is mapped to multiple elements $x$ in a domaine $X$ A function $f$ with domain $X$ and codomain $Y$ is surjective if for every $y$ in $Y$ there exists at least one $x$ in $X$ with $f(x)=y$. But what if there is a case where every $x$ in $X$ is connected to multiple elements in $y$. How would one describe such a case?
representation of the case
 A: Let $X:=\mathbb{R}\setminus\{0\}$, $Y:=(0,\infty)$, and $f(x):=x^2$ for all $x\in X$.
A: The title of the question describes a function which is not one-to-one (or injective).
This addresses the body of the question: A relation in which there is more than one $y$ for each $x$ is not a function.   I have heard about distributions being generalisations of functions.  On the other hand,  in complex analysis,  Riemann surfaces deal with such objects.  
A function maps, by definition,  each element of the domain to a single element of the codomain. 
A: This is called a multivalued function. Like a function, it's a set of pairs from $X \times Y$, such that every domain element $x$ has at least one "value" associated with it, so $$\forall x \in X: \exists y \in Y: (x,y) \in f$$
A standard (single-valued) function also has he property that images of a point are unique:
$$\forall x \in X: \forall y,y' \in Y: ((x,y) \in f \land (x,y') \in f) \implies y=y'$$
You seem to be interested in the former variety. You could model them as classical single-valued functions with domain $X$ and codomain $\mathscr{P}(Y)\setminus \{\emptyset\}$, sending $x$ to its (non-empty) set of all possible values. That set then be seen as its unique function image. This does change the codomain of course.
