Function Arithmetic & Composition Calculator

None Result only.

Summary Method pills.

Detailed Full accordion: definition → substitution → sample table → domain → commutativity check.

f+g — Sum

f(x)+g(x), zeros where f=−g, domain D(f)∩D(g)

f-g — Difference

f(x)−g(x), zeros where f=g (intersections), NOT commutative

f·g — Product

f(x)·g(x), zeros: f=0 OR g=0, commutative

f/g — Quotient

f(x)/g(x), exclude g=0, horizontal asymptote

f∘g — Composition

f(g(x)), NOT commutative, domain two-step check

f∘g∘h — Triple

f(g(h(x))), inside-out, associative but not commutative

D(f) — Domain

Parity, individual D(f), D(g), D(f±g), D(f/g), D(f∘g)

h=f∘g — Decompose

Write h as f(g(x)), natural inner g auto-detected

Sum \((f+g)(x)=f(x)+g(x)\)

Difference \((f-g)(x)=f(x)-g(x)\)

Product \((fg)(x)=f(x)\cdot g(x)\)

Quotient \((f/g)(x)=f(x)/g(x),\;g\neq 0\)

Composition \((f\circ g)(x)=f(g(x))\)

Triple \((f\circ g\circ h)=f(g(h(x)))\)

D(f+g) \(D_f\cap D_g\)

D(f/g) \(D_f\cap D_g\setminus\{g=0\}\)

D(f∘g) \(\{x\in D_g:\,g(x)\in D_f\}\)

Not commutative \(f\circ g\neq g\circ f\text{ in general}\)

Associative \((f\circ g)\circ h=f\circ(g\circ h)\)

Zero Product \(fg=0\Leftrightarrow f=0\text{ or }g=0\)

Even function \(f(-x)=f(x)\;(y\text{-axis symmetry})\)

Odd function \(f(-x)=-f(x)\;(\text{origin symmetry})\)

Identity \(f\circ\mathrm{id}=\mathrm{id}\circ f=f\)

Chain Rule \((f\circ g)'=f'(g(x))\cdot g'(x)\)

f∘gFunction Arithmetic & Composition