Why do we focus so much in math on functions (as a subclass of relations)?

Question

Why is it that math so focuses on the subclass of relations known as functions? I.e. why is it so useful for us in nearly all branches of mathematics to focus on relations which are left-total and left-unique? Left- (or even right-) totality seem to be intuitive, since if an element doesn't appear in the domain, we might throw it out. But why left-uniqueness?

I'm looking for something like a "moral explanation" of why they would be the most useful subclass of relations.

My apologies if this is a previous question; I looked and didn't find much.

Answer 1

A function models a deterministic computation: if you put in $x$, you always get out the same result, $f(x)$, hence the left-uniqueness.

The asymmetry of the definition (left uniqueness rather than right uniqueness) is because the left side models the input and the right side models the output, and the input is logically prior to the output. If you know the input, you can determine the output, but you can't (in general) do the reverse. The function $f:x\mapsto x^2$ means that if you put in 17 you get out 289. But it makes no sense at all to ask what you get out before specifying what you put in.

Answer 2

I would say functions model our intuitive notion of directionality (this is a similar, though still slightly different, answer than MJD's). After all the notation for functions is $$X\xrightarrow{\;\;f\;\;}Y$$ and not $$X\overset{f}{\;\;\mathrel{\star\small\vdots\!\smallsmile\!\wr}\;\;}Y$$