Breaking Apart

Composition builds a function by putting one rule inside another. Breaking apart runs that backwards: you look at a rule and find two simpler rules that make it. Take h(x) = (2x + 1)^3. To work out h(1) you would first compute 2 times 1 plus 1, then cube the result.

Two steps, so two functions. The inner one, done first, is g(x) = 2x + 1. The outer one, done to its output, is f(x) = x^3. Then h(x) = f(g(x)). Check by composing: putting 2x + 1 in place of x in x^3 gives back (2x + 1)^3.

The order of operations tells you which piece is inside: whatever happens to x first is the inner function, and the last step is the outer one. It is the same as splitting one long line of code into two small helper functions you already understand.