A topic in the Open Knowledge Graph — a free, open map of 15,290 topics and the order to learn them in.
The composition of f and g, written (f o g)(x) = f(g(x)), means applying g first, then applying f to the result. Composition is not commutative: f(g(x)) is generally different from g(f(x)). The domain of f o g is all x in the domain of g such that g(x) is in the domain of f. Composition is the foundation for understanding inverse functions, the chain rule in calculus, and function decomposition.
Start by evaluating compositions at specific values: if f(x) = x2 and g(x) = x+1, find f(g(3)) step by step. Then find the composition formula f(g(x)). Practice both f(g(x)) and g(f(x)) to demonstrate non-commutativity. Discuss domain restrictions. Decompose complex functions into compositions of simpler ones.
Function notation like f(x) means "apply rule f to input x and get an output." Composition of functions takes this one step further: the output of one function becomes the input of another. When you write f(g(x)), you are chaining two rules together — g processes x first, then f processes whatever g produced.
A concrete example makes this tangible. Let g(x) = x + 1 (add one) and f(x) = x² (square). Then f(g(3)) means: first apply g to 3 — you get 4 — then apply f to 4 — you get 16. The notation tells you the order: the innermost function goes first. Always work from inside out. If you applied f first you would get f(3) = 9, then g(9) = 10, which is g(f(3)) — an entirely different computation.
This non-commutativity — f(g(x)) ≠ g(f(x)) in general — is the most important property to internalize. In arithmetic, 3 + 4 = 4 + 3 and 3 × 4 = 4 × 3. Function composition breaks this symmetry. Think of it like getting dressed: socks then shoes gives a very different result from shoes then socks, even though the individual steps are the same.
To find the composed formula f(g(x)) in general, substitute the entire expression for g(x) wherever x appears in f. With f(x) = 2x + 1 and g(x) = x², write f(g(x)) = 2(x²) + 1 = 2x² + 1. Notice that f(x) · g(x) = (2x+1)(x²) = 2x³ + x² — a completely different expression. Composition is substitution, not multiplication.
Composition is the hidden structure inside many formulas you will encounter. In calculus, the chain rule exists precisely to differentiate composed functions. When you see a complex expression like sin(x²) or e3x+1, you are looking at a composition — one function wrapped inside another. Learning to decompose and recompose functions is a foundational skill for everything that follows.