A composite function is what you get when the output of one function becomes the input of another. In symbols, \(h(x)=f(g(x))\), read as "f of g of x."
- The inner function \(g\) receives the original input \(x\) and produces \(g(x)\).
- The outer function \(f\) receives that output and produces \(f(g(x))\).
So the expression is not a product and not a sum — it is a chain of two processing steps, in a fixed order. The inner step always runs first, because the outer function has nothing to act on until the inner function has produced a value.
Example: for \(h(x)=\sin(3x^2)\), the inner function is \(g(x)=3x^2\) and the outer function is \(f(u)=\sin u\). An input of \(x=1\) first produces \(3\), and then \(\sin 3\).
This ordering matters for everything that follows. When you later differentiate, the derivative structure will mirror this same chain: the outer function's rate of change is scaled by the inner function's rate of change.