How do you get a function of a function?

Answer 1

A function is a mapping from one set of numbers (domain) to another (range). The mapping need not be linear: it can be any mathematical function. That is, for every number in the domain the function provides a rule which allows you to calculate another number.

If, then, you devise another function which is a mapping from the range of the first function to some other set, you have a function of a function.

For example, suppose the first function, f, is "add 1" and the second function, g, is "square the number."

Then the function

g of f = g[f(x)] = g[x+1] = [x+1]2 = x2 + 2x + 1

however, note that

f of g = f[g(x)] = f[x2] = x2 + 1

This illustrates that f of g is not the same as g of f.