Evaluate and build composite functions algebraically, and determine the domain of a composite function.
Composition feeds the entire output of one function into another: (f o g)(x) means "do g first, then feed that result into f," written f(g(x)). Order matters enormously -- f(g(x)) and g(f(x)) are generally two completely different functions, so always work from the inside out.
The domain of a composite function f(g(x)) needs two things to hold at once: x must be in the domain of g (the inner function), AND g(x) itself must land in the domain of f (the outer function). Missing that second condition is the single most common composition mistake -- a value can be perfectly fine for g and still break f once you see what g(x) actually equals.
Mastery looks like: They correctly build a composite function, evaluate it either by formula or inside-out, and find its full domain by checking both the inner function and the outer function's requirement.
Common stumbles: Substituting into the wrong function first (outside-in instead of inside-out); checking only the inner function's domain and missing a restriction the outer function adds.