Function Composition Calculator — f(g(x)) and g(f(x))

How it works

(f ∘ g)(x) = f(g(x)): replace every x in f with the whole expression g(x), then expand and collect like terms · (g ∘ f)(x) = g(f(x)) · deg f(g(x)) = deg f × deg g

Composition feeds the output of one function into another: in f(g(x)) the inner function g acts first and f acts on its result. The calculator substitutes g(x) into f symbolically using exact fractions, expands the powers and collects like terms, and does the same the other way round. Composition is usually not commutative, so f(g(x)) and g(f(x)) are generally different polynomials. The value at a chosen x is found by evaluating the inner function first, which is how textbooks (OpenStax Precalculus §1.4) teach it.

Worked example

f(x) = x² + 1 and g(x) = 2x − 3. f(g(x)) = (2x − 3)² + 1 = 4x² − 12x + 9 + 1 = 4x² − 12x + 10. g(f(x)) = 2(x² + 1) − 3 = 2x² − 1. At x = 2: g(2) = 1, so f(g(2)) = f(1) = 2; f(2) = 5, so g(f(2)) = g(5) = 7. The two compositions differ, so the order matters.

Sources: OpenStax Precalculus 2e — 1.4 Composition of Functions · Khan Academy — Composite functions

Frequently asked questions