Decomposing Composite Functions

Pre-Calculus · Axiom Academy

EXAMPLE Decomposing Composite Functions Breaking a complex function apart into a simpler inner and outer piece Given the composite function h(x) = (3x + 2)^5 , find functions f and g such that h(x) = f(g(x)) — that is, identify the inner function g (what's inside) and the outer function f (what is done to it). Nice work — you decomposed h(x) = (3x + 2)^5 into g(x) = 3x + 2 and f(u) = u^5 , so that f(g(x)) = h(x) . Look for the nested structure: spot the outermost operation, then read off what it acts on. Inner function g(x) : the expression sitting inside the main operation — here, 3x + 2 . Outer function f(u) : the operation applied to that inner piece — here, raising to the 5th power, f(u) = u^5 . Always verify: compose f(g(x)) and confirm you get the original — f(3x + 2) = (3x + 2)^5 = h(x) . Not unique: a composite can have more than one valid decomposition. Decomposing functions this way is exactly what sets up the Chain Rule in calculus: .

This is the written version of the interactive lesson above. See the full Pre-Calculus course.