Function Composition Calculator

About this calculator

Function composition (f∘g)(x) means feeding the output of one function directly into another: first compute g(x), then plug that result into f. For two linear functions f(x) = a₁x + b₁ and g(x) = a₂x + b₂, the composite works out to (f∘g)(x) = a₁(a₂x + b₂) + b₁ = (a₁a₂)x + (a₁b₂ + b₁), which is itself a linear function with slope a₁a₂ and intercept a₁b₂ + b₁.

Composition is a core idea in algebra and precalculus courses because it generalizes far beyond lines — the chain rule in calculus is essentially the derivative version of composing functions, and chaining together simple transformations (like scaling, then shifting, then scaling again) to build a more complex one is a technique used throughout math, computer graphics, and signal processing.

Enter the slope and intercept of two linear functions along with a point x, and the calculator computes g(x), then f(g(x)), plus the overall slope and intercept of the composite function.