Why Is a Function Defined As Having Only One Y-Value Output? I know that it is defined such that there is no more than one y-value output for any given x-value input, but I"m wondering WHY it is defined that way? Why can't we apply everything we know about functions to equations that have more than one y-value output for any given x-value input? 

In short, what is the benefit or reason to define a function this way?  At the same time, a little historical context would also be appreciated (i.e., when the concept originated and why).

Thanks,

Moshe
Strictly speaking, a &quot;well-defined&quot; function associates one, and only one, output to any particular input. A function would not be well-defined if say $y = f(x)$ such that $y$ can take on any number of values at any particular input. Suppose the input $\;x = a\;$ outputs more than one distinct value, so that $y = f(a) \in\{y_1, y_2, ..., y_k, ...\} $. We'd never be able to say what, precisely, $y$ is when $x = a$.
If $y = f(a) = b$, and $y = f(a) = c$, and $\,b\ne c$, then the value of the function  $\,\,f(x) = y\,$ at $\,a\,$ is ambiguous: $y$ would not specify any particular value at $a$. That is, its value is not well-defined at $a$, and perhaps not well-defined at other inputs, as well. Nor can we say much about the behavior of a function at a particular value, if it can take on many values at a given point.
E.g. How would we define continuity of, say, a real-valued multifunction?
The strict definition of a function, in terms of &quot;outputing&quot; exactly one value for any given input is really no more than an attempt to keeping functions well-defined, and thus properties of functions well-defined.
But you'd might like to explore the following:
See this entry on multi-valued &quot;functions&quot;:

A multivalued function (shortly: multifunction, other names: many-valued function, set-valued function, set-valued map, multi-valued map, multimap, correspondence, carrier) is a left-total relation; that is, every input is associated with at least one output.
The term &quot;multivalued function&quot; is, therefore, a misnomer because functions are single-valued. Multivalued functions often arise from functions which are not injective. Such functions do not have an inverse function, but they do have an inverse relation. The multivalued function corresponds to this inverse relation. [bold-face mine]
Obviously this limitation of defining a function f(x) as giving only ONE value of y is specific to the English (or American?) Maths.
In French, we consider y^2 = x as a function, we call it horizontal parabole...
See http://www.alloprof.qc.ca/BV/pages/m1330.aspx
I would be tempted to use pejorative language toward English (or AMerican?) Maths!