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
Best Answer
Strictly speaking, a "well-defined" 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 "outputing" 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 "functions":