As everybody else pointed out, functions have that property by definition; herp derp
I think what you're trying to ask is, why not consider a more general class of "functions"? If you lrn2 wiki, then you'd find:
http://en.wikipedia.org/wiki/Multivalued_function
http://en.wikipedia.org/wiki/Relation_(mathematics)
Math is a lot more general than you think; just because it's not taught doesn't mean it's not considered. Multivalued "functions" most often crop up as inverses (inverse functions are often very important) of noninjective (not 1-1) functions (eg sin, x^2...). It just so happens that such entities are a hueg pain in the ass. If you've ever worked with complex logs, you'd know what I mean.
This is also important in computers, where determinism is important. You want the computer to spit out 1 when you ask for the 4th root of 1. If it sometimes spits out -1, or +-i, your code is bound to break.