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

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
$$X\overset{f}{\;\;\mathrel{\star\small\vdots\!\smallsmile\!\wr}\;\;}Y$$
