Q: Why do we care about functions?
A: Physical concepts are relationships between experimentally measurable quantities. By introducing the concept of a "function" we can be more precise about what we mean by "relationship." Describing specific examples of relationships can often be involved and time-consuming. The concept of a function lets us assign short names by which we can efficiently refer to particular discussions of quantitative relationships, which include specification of the contexts in which these relationships are described.
When we agree to define "function" and "composition of function" we generate an abstract view of part of the process of combining quantitative hypotheses and firmly deducing the quantitative hypotheses they imply.
There is also a discussion (http://mathforum.org/library/drmath/view/62559.html) at the MathForum.
