Function composition

In mathematics, the composition operator $\circ$ takes two functions, $f$ and $g$ , and returns a new function $h(x):=(g\circ f)(x)=g(f(x))$ . Thus, the function $g$ is applied after applying $f$ to $x$ . $(g\circ f)$ is pronounced "the composition of $g$ and $f$ ".^[1]

Reverse composition applies the operation in the opposite order, applying $f$ first and $g$ second. Intuitively, reverse composition is a chaining process in which the output of function $f$ feeds the input of function $g$ .

The composition of functions is a special case of the composition of relations, sometimes also denoted by $\circ$ . As a result, all properties of composition of relations are true of composition of functions,^[2] such as associativity.

^ "Composition of Functions". nool.ontariotechu.ca. Retrieved 2025-02-07.
^ Cite error: The named reference Velleman_2006 was invoked but never defined (see the help page).

[1] "Composition of Functions". nool.ontariotechu.ca. Retrieved 2025-02-07.

[Velleman_2006-2] Cite error: The named reference Velleman_2006 was invoked but never defined (see the help page).

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Function
$x \mapsto f (x)$
History of the function concept
Types by domain and codomain
$X \to 𝔹$ $𝔹 \to X$ $𝔹 n \to X$ $X \to ℤ$ $ℤ \to X$ $X \to ℝ$ $ℝ \to X$ $ℝ n \to X$ $X \to ℂ$ $ℂ \to X$ $ℂ n \to X$
Classes/properties
Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective Bijective
Constructions
Restriction λ Inverse
Generalizations
Relation (Binary relation) Set-valued Multivalued Partial Implicit Space Higher-order Morphism Functor
List of specific functions