Why do we use function symbol instead of defining it in terms of predicates? Set theory defines function as a kind of relations/predicates. If so, then why we introduce function symbol instead of introducing $n$-ary function as a $(n+1)$-ary predicate symbol?
 A: I guess the "functional way" is easier to teach and hard to get rid of later (and not necessary enough). Most people feel well with a function being a "machine" or "process", something goes in and something else comes out. This is how it appears in reality, e.g. when using a calculator. We not really relate the input and the output to each other. 
Like many other set theoretic constructions, this should probably just be seen as a foundational thing. Or would you say that you see


*

*natural numbers as specific finite sets,

*whole numbers as (equivalence classes of) pairs of natural numbers,

*real numbers as sets of Cauchy sequences?


In the end, what is important is the concept of what a function is instead of how it is constructed in the currently most used foundational system. There are other ways do build math, e.g. Category theory. And if you want to build a function in there, you better know what the idea of a function is and not what its specific construction is in any other foundational system.
However, that the specific notation $f(x)=y$ is seen as "better" as $(x,y)\in f$ is purely educational to some degree. In the end, also the latter notation can describe a mapping process. I just think that the first way better communicates the distinction between $x$ and $y$ and that everywhere $f(x)$ can be substituted by $y$.
A: There are many different possible answers to this question. 
For start one could argue that there are different foundational systems, as Martin Löf type theory, which has functions as a primitive notion (as opposed to a defined one).
Another possible reason, a little more technical, is that having function symbols allows to build term models which is one of the common technique in model theory for building up models for theories.
Still a technical answer: adding function symbols is a nice way to extend the theory in such a way that we have quantifier elimination, a nice property of theories that comes in handy for proving stuff like completeness and other properties of a theory.
These are just few reason why having function symbols is important, I believe other with more knowledge than me in logic and specifically in model theory can give you many more interesting reasons.
