Function composition

In mathematics, function composition is an operation $\circ$ that takes two functions $f$ and $g$ , and produces a function $h = g \circ f$ such that $h (x) = g (f (x))$ . In this operation, the function $g$ is applied to the result of applying the function $f$ to $x$ . That is, the functions $f : X \to Y$ and $g : Y \to Z$ are composed to yield a function that maps $x$ in domain $X$ to $g (f (x))$ in codomain $Z$ . Intuitively, if $z$ is a function of $y$ , and $y$ is a function of $x$ , then $z$ is a function of $x$ . The resulting composite function is denoted $g \circ f : X \to Z$ , defined by $(g \circ f)(x) = g (f (x))$ for all $x$ in $X$ .^{[nb 1]}

The notation $g \circ f$ is read as " $g$ of $f$ ", " $g$ after $f$ ", " $g$ circle $f$ ", " $g$ round $f$ ", " $g$ about $f$ ", " $g$ composed with $f$ ", " $g$ following $f$ ", " $f$ then $g$ ", or " $g$ on $f$ ", or "the composition of $g$ and $f$ ". Intuitively, composing functions 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,^[1] such as the property of associativity.

Composition of functions is different from multiplication of functions (if defined at all), and has some quite different properties; in particular, composition of functions is not commutative.^[2]

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[3] "3.4: Composition of Functions". Mathematics LibreTexts. 2020-01-16. Retrieved 2020-08-28.

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Function
$x \mapsto f (x)$
History of the function concept
Examples of domains and codomains
$X$ → $\mathbb {B}$ , $\mathbb {B}$ → $X$ , $\mathbb {B} ^{n}$ → $X$ $X$ → $\mathbb {Z}$ , $\mathbb {Z}$ → $X$ $X$ → $\mathbb {R}$ , $\mathbb {R}$ → $X$ , $\mathbb {R} ^{n}$ → $X$ $X$ → $\mathbb {C}$ , $\mathbb {C}$ → $X$ , $\mathbb {C} ^{n}$ → $X$
Classes/properties
Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective Bijective
Constructions
Restriction Composition λ Inverse
Generalizations
Binary relation Partial Multivalued Implicit Space
v t e