What is a function Update Part 1

What's a function? Part 1 Updated 8/10/2013. Here we discuss the abstract definition of a function. We examine the notion of a morphism as a function that preserves some mathematical structure in the domain of f as it maps elements of f to the range of f preserving the structure in the Range of f. We discuss functions on the real numbers and the powerful concept of restrictions on the domain and range of a function. For example, the square root of a real number is a function having two branches (so it's really two distinct functions) having its domain restricted to the non-negative real numbers. An example of a branch point is given for the functions f(x) = +/- SQRT(x). We introduce the powerful notion of mapping sets to sets. We define the image of a set V in the domain of f as the set G in the range of f such that f(V)=G. G is the image of V under f.