We know function notation: f(x) means "run the rule f on the input x". A composite function simply chains two rules together — you do one, then feed its output straight into the other.

We write the composite of f and g as fg(x), or more clearly as f(g(x)). The golden rule is to work from the inside out:

The function nearest the x goes first. So in f(g(x)) the inner function g runs first, and whatever it produces becomes the input to f.

Two machines in series

Think of it as two function machines bolted together. A number drops into the first machine g; its output slides along and drops into the second machine f; what comes out the far end is f(g(x)). Press play to watch a number travel through both.

Building the new rule

We can also combine the two rules into one formula. With f(x) = x + 1 and g(x) = 2x, we substitute the whole rule for g wherever f expects its input:

Now check a number against the machines above: with x = 3, g(3) = 6 and then f(6) = 7, which matches 2(3) + 1 = 7.

Order matters

Swapping the order usually gives a different rule. Doing f first and then g is the composite g(f(x)):

That is not the same as f(g(x)) = 2x + 1. Composing functions is a kind of careful substitution, and the order you apply the machines changes the answer.

Khan Academy introduces composing functions here: Intro to composing functions.