Decomposing Functions

Pre-Calculus · Axiom Academy

Take a composite apart: spot the inner function done first, the outer function done to it, and check that f(g(x)) rebuilds the original. Decomposing means finding two simpler functions f and g so that the original equals f(g(x)) — you're breaking one complicated machine back into its building blocks. Take h(x) = (x+1)^3 . Watch an input flow through it: first it goes into the inner machine g(x)=x+1 , and that result u is handed to the outer machine f(u)=u^3 . 2. The Strategy — Find the Core To decompose, hunt for the core : an expression nested inside another operation. Ask "what happens first?" — that's g . Then ask "what happens to it?" — that's f . For , the square root is the last thing you'd do, so everything underneath it, x^2+1 , is the core. Look for a core — an expression nested inside another operation. Let g(x) = the core — the inner part that gets processed first. Let f(u) = what happens to the core — rewrite the rest with the core replaced by u . Verify that f(g(x)) returns the original function. An expression has layers, innermost to outermost. For : the innermost is x^2 , then +1 wraps it into the core x^2+1 , then wraps the core. The outermost wrapper is f ; its contents are g . A decomposition is only correct if it rebuilds the original. To check, substitute the whole inner function into the outer one — replace every u in f(u) with g(x) — and confirm you land back on h(x) . With f(u)=u^5 and g(x)=2x-3 , plugging in gives f(g(x)) = (2x-3)^5 , exactly h(x) .

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