Domain of a Composite, and Decomposing

A composite function f(g(x)) runs x through g first, then feeds that result into f. For x to be allowed, two things must hold: x has to be a legal input for g, and the output g(x) has to be a legal input for f. That second condition is the one students forget. It's why a composite can have a smaller domain than f or g alone — g might accept some x just fine, but hand f an output f can't use, like a negative number under a square root or a zero in a denominator.

Mechanically, you work from the outside in but check from the inside out. Find g's own domain restrictions first, then write down what f needs as input, then translate that requirement back onto g(x), and solve for x. The algebra usually turns into an inequality or equation involving g's formula, since you're asking which x-values make g(x) satisfy f's requirement.

Decomposing is this process backwards: given one messy formula, find an inner function g(x) and outer function f(x) so that f(g(x)) matches it. Look for the last operation applied — that's usually the outer function — and whatever it's being applied to is the inner one. This skill connects directly to transformations: shifting a graph horizontally by k is really composing f with g(x) = x - k, since you're changing what gets fed into f before f ever acts. Seeing shifts as composition ties this topic straight back to how transformed graphs behave.