composite function

A composite function is what you get by chaining two functions together: do one, then feed its answer straight into the other. It is an assembly line. The first machine turns a log into planks; the second machine turns planks into a chair. Run them back to back and you have a single combined process that turns logs into chairs.

If g acts first and f acts second, the composite is written (f o g)(x) = f(g(x)), read 'f of g of x'. You evaluate from the inside out: compute g(x), then apply f to that result. For example, with g(x) = x + 1 and f(u) = u^2, the composite (f o g)(x) = (x + 1)^2. The inner function's output must be a legal input for the outer function, so the domain of the composite can be narrower than g's own domain.

Order matters: in general f(g(x)) is not the same as g(f(x)). With the functions above, g(f(x)) = x^2 + 1, which is plainly different from (x + 1)^2. This is more than bookkeeping — recognizing a function as an inside wrapped in an outside is exactly the setup the chain rule needs in order to differentiate it.