Importance of function

[Christ]

In many branches of mathematics ,concept of function appears. It seems to be naturally. But I was searching more recondite reasons for this matter.
Many of applications and also facts deal with mathematics, may expose with
respect to philosophy by philosophers .Time ,motion ,point ,line ,energy, etc are
a few examples .

Above mentioned matters may regard as cause and effect ,matter and form ,and
such dual concept which are important in philosophy . On the other hand a function may express such concepts .

A function usually contains two sets , Domain and co-domain .These are
ordered as first and second . That implies precession of cause. Each member of domain relates to only one member of co-domain that may be similar to effect
uniqueness .

Such ideas may be reason of function importance. Considering function as this point of view may help us to get more insight to this concept .

[Marabod]

I can suggest some further reading on the topic, this would be a school textbook on Algebra. An impressing thing, I would tell ya!

[Keen]

Although you gave a very good definition of a function, I don't see quite your point. The only link between function and causality I can see is that causes preceede consequences, like the antecedent preceedes the image of a function. I don't think this a sufficient reason for your interpretation of function.
For me a function is some sort of recipee. It is the way of writing how to transform an object into another without actually doing so. In mathematics the important things are not the objects but the structures and the links between them, the functions are fundamental in this science.