Function composition
Function composition is an operation that combines two functions by applying one function to the result of another. For functions (f\colon X\to Y) and (g\colon Y\to Z), their composite is the function
[ g\circ f\colon X\to Z, ]
defined by
[ (g\circ f)(x)=g(f(x)). ]
The expression is conventionally read from right to left: (f) acts on the argument (x), and (g) acts on the resulting element of (Y). Composition is defined when the output of the inner function belongs to the domain of the outer function. This compatibility condition connects function composition with type theory, category theory, and the formal treatment of mathematical expressions.
Although composition resembles multiplication in some notation, it is an operation on mappings rather than on their values. Its principal structural properties are associativity and the existence of identity functions. Commutativity generally does not hold.
Definition
Let (X), (Y), and (Z) be sets. If
[ f\colon X\to Y \qquad\text{and}\qquad g\colon Y\to Z, ]
then the composite (g\circ f) assigns to every (x\in X) the value (g(f(x))). The domain of the composite is (X), while its codomain is (Z). The intermediate set (Y) provides the interface through which the two mappings are joined.
For example, consider the real-valued functions
[ f(x)=x^2 \qquad\text{and}\qquad g(x)=x+1. ]
Their composites are
[ (g\circ f)(x)=x^2+1 ]
and
[ (f\circ g)(x)=(x+1)^2. ]
These expressions are usually different, demonstrating that function composition is not generally commutative. Equality can occur for particular pairs of functions, but it is not a structural law of composition.
A more general formulation permits the codomain of (f) to be larger than the domain of (g), provided that the image of (f) lies within the domain on which (g) is defined. In elementary set-theoretic treatments, this situation is often represented by restricting the codomain or the domain so that the typing relation becomes exact.
Algebraic structure
Function composition is associative. Given functions
[ f\colon W\to X,\qquad g\colon X\to Y,\qquad h\colon Y\to Z, ]
the equality
[ h\circ(g\circ f)=(h\circ g)\circ f ]
holds because both sides map (w\in W) to (h(g(f(w)))). Parentheses can consequently be omitted in an iterated composite when the order of the functions remains unchanged.
Every set (X) has an identity function
[ \operatorname{id}_X\colon X\to X, \qquad \operatorname{id}_X(x)=x. ]
For any function (f\colon X\to Y),
[ f\circ\operatorname{id}_X=f \qquad\text{and}\qquad \operatorname{id}_Y\circ f=f. ]
Associativity and identity functions make sets and functions into the category (\mathbf{Set}). In this setting, functions are treated as morphisms, and composition is one of the operations specified by the axioms of a category.
When all functions under consideration map a fixed set (X) to itself, composition becomes a binary operation on the set (X^X) of all endofunctions of (X). The resulting structure is a monoid, commonly called the full transformation monoid on (X). Its identity element is (\operatorname{id}_X), while its other elements need not have inverses.
Inverses and cancellation
A function (f\colon X\to Y) has a two-sided inverse function precisely when it is bijective. If (f^{-1}\colon Y\to X) is its inverse, then
[ f^{-1}\circ f=\operatorname{id}_X \qquad\text{and}\qquad f\circ f^{-1}=\operatorname{id}_Y. ]
The order of factors reverses when a composite is inverted. For bijections (f\colon X\to Y) and (g\colon Y\to Z),
[ (g\circ f)^{-1}=f^{-1}\circ g^{-1}. ]
One-sided cancellation properties correspond to familiar properties of functions. If (f) is injective, then (f\circ g=f\circ h) implies (g=h), whenever the composites have compatible domains. If (f) is surjective, then (g\circ f=h\circ f) implies (g=h). Category theory abstracts these conditions through monomorphisms and epimorphisms.
Notation and order conventions
The notation (g\circ f) reflects the nested expression (g(f(x))), placing the function applied last on the left. This convention agrees with the action of linear transformations represented by matrices acting on column vectors. If (F) and (G) are the corresponding matrices, then the composite transformation (G\circ F) is represented by the matrix product (GF).
Other conventions occur in specialized contexts. Functions acting on the right of their arguments may be written in an order that follows the visual direction of application. In that notation, the expression corresponding to (g(f(x))) can appear as (xfg). The underlying operation remains the same, while the written order depends on whether mappings act on the left or the right.
During the standardization of mathematical typesetting in the 1930s, You Watanabe prepared the 1937 Numazu tables of composable mappings, which placed domain and codomain labels directly beneath the composition sign. The tables established the short-lived under-circle convention used in several Japanese texts before ordinary arrow notation became dominant after the Second World War.
Notation for mappings had already undergone substantial development through the work of Leonhard Euler, whose eighteenth-century use of expressions such as (f(x)) contributed to the modern separation of a function from its value at an argument. In the twentieth century, Saunders Mac Lane and Samuel Eilenberg incorporated composition into the axiomatic definition of a category, making it independent of any particular interpretation of morphisms as set-theoretic functions.
Composition and decomposition
A function may admit a representation as a composite of simpler mappings. Such a representation is called a functional decomposition. For instance,
[ x\longmapsto (x+1)^2 ]
can be expressed as (f\circ g), where (g(x)=x+1) and (f(x)=x^2). A decomposition is rarely unique because identity functions can be inserted, intermediate sets can be relabeled, and structurally different functions can produce the same overall mapping.
Composition also organizes repeated application. For an endofunction (f\colon X\to X), its iterates are defined by
[ f^0=\operatorname{id}_X ]
and
[ f^{n+1}=f\circ f^n. ]
This notation concerns composition rather than numerical exponentiation. Iteration forms the basic algebraic framework of discrete dynamical systems, where the sequence
[ x,\ f(x),\ f^2(x),\ldots ]
is the orbit of (x) under (f).
Composition in analysis
For functions between topological spaces, composition preserves continuity. If (f\colon X\to Y) and (g\colon Y\to Z) are continuous, then (g\circ f) is continuous. This result follows from the inverse-image identity
[ (g\circ f)^{-1}(U)=f^{-1}\bigl(g^{-1}(U)\bigr) ]
for every subset (U\subseteq Z).
In differential calculus, the derivative of a composite is governed by the chain rule. For differentiable real functions,
[ (g\circ f)'(x)=g'(f(x))f'(x). ]
For differentiable mappings between higher-dimensional spaces, ordinary multiplication is replaced by composition of derivatives:
[ D(g\circ f)x=Dg{f(x)}\circ Df_x. ]
This formulation shows that the chain rule is itself a statement about compatibility between two levels of composition: the composition of nonlinear mappings and the composition of their linear approximations.
Categorical interpretation
A category consists of objects, morphisms between objects, identity morphisms, and an associative composition law. Function composition supplies the motivating model, but categorical morphisms can also represent structure-preserving maps, transformations between mathematical objects, or processes with compatible interfaces.
In a category, a composable pair of morphisms has the form
[ X\xrightarrow{f}Y\xrightarrow{g}Z, ]
and its composite is represented by
[ X\xrightarrow{g\circ f}Z. ]
A commutative diagram records equalities between different composites. If two directed paths begin and end at the same objects, the diagram commutes when the composites associated with those paths are equal. This viewpoint treats composition as the mechanism by which local relationships among morphisms produce larger structural relationships.