Seconding this. I've had this discussion a number of times and even when people initially reject the "relation" definition, if you push them with the Socratic Method eventually they get to a point where they see it really is the only good way to define functions, regardless of how "laymen unfriendly" relations seem.

The 'machine' "definition" perpetuates as an idiom, with teachers who don't know any better teaching it so their students don't know any better, and it does great damage to math education.