In mathematics, function composition is the pointwise application of one function to the
result of another to produce a third function. For instance, the functions f : X → Y
and g : Y → Z can be composed to yield a function which maps x in X to g(f(x)) in
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 ∘ f : X → Z, defined by(x)
= g(f(x)) for all x in X. The notation g ∘ f is read as "g circle f ", or "g round f ", or
"g composed with f ", "g after f ", "g following f ", or "g of f ". Intuitively, composing
two functions is a chaining process in which the output of the first function becomes the
input of the second function. The composition of functions is just a particularization
of the composition of relations, so all properties of the latter operation also transfer to the
composition of functions. The composition of function has some additional properties
however.
Examples
Composition of functions on a finite set: If f = {(1,3),,,}, and g = {(1,5),,,,,}, then
g ∘ f = {(1,4),,,}. Composition of functions on an infinite set:
If f: R → R is given by f(x) = 2x + 4 and g: R → R is given by g(x) = x3, then:
(f ∘ g)(x) = f(g(x)) = f(x3) = 2x3 + 4, and
(g ∘ f)(x) = g(f(x)) = g(2x + 4) =3. If an airplane's elevation at time t is given
by the function h(t), and the oxygen concentration at elevation x is given by the function c(x),
then(t) describes the oxygen concentration around the plane at time t.
Properties The composition of functions is always associative—a
property inherited from the composition of relations. That is, if f, g, and h are three
functions with suitably chosen domains and codomains, then f ∘ = ∘ h, where the parentheses
serve to indicate that composition is to be performed first for the parenthesized functions.
Since there is no distinction between the choices of placement of parentheses, they
may be left off without causing any ambiguity. In a strict sense, the composition g ∘ f
can be built only if f's codomain equals g's domain; in a wider sense it is sufficient
that the former is a subset of the latter. Moreover, it is often convenient to tacitly
restrict f's domain such that f produces only values in g's domain; for example, the composition
g ∘ f of the functions f : ℝ → = 9 − x2 and g : [0,+∞) → ℝ defined
by g(x) = √x can be defined on the interval [−3,+3].
The functions g and f are said to commute with each other if g ∘ f = f ∘ g.
In general, composition of functions will not be commutative. Commutativity is a special
property, attained only by particular functions, and often in special circumstances. For example,
|x| + 3 = |x + 3| only when x ≥ 0. The picture shows another example.
The composition of one-to-one functions is always one-to-one. Similarly, the composition
of two onto functions is always onto. It follows that composition of two bijections is also
a bijection. The inverse function of a composition has the property that−1 =.
Derivatives of compositions involving differentiable functions can be found using the chain rule.
Higher derivatives of such functions are given by Faà di Bruno's formula.
Composition monoids
Suppose one has two functions f: X → X, g: X → X having the same domain and codomain;
these are often called transformations. Then one can form chains of transformations composed
together, such as f ∘ f ∘ g ∘ f. Such chains have the algebraic structure of a monoid,
called a transformation monoid or composition monoid. In general, transformation monoids
can have remarkably complicated structure. One particular notable example is the de Rham
curve. The set of all functions f: X → X is called the full transformation semigroup
or symmetric semigroup on X.
If the transformation are bijective, then the set of all possible combinations of these
functions forms a transformation group; and one says that the group is generated by these
functions. A fundamental result in group theory, Cayley's theorem, essentially says that any
group is in fact just a group of permutations. The set of all bijective functions f: X → X
forms a group with respect to the composition operator. This is the symmetric group, also
sometimes called the composition group. In the symmetric semigroup one also finds
a weaker, non-unique notion of inverse because the symmetric semigroup is a regular semigroup.
Functional powers
If Y ⊆ X, then f: X→Y may compose with itself; this is sometimes denoted as f 2.
That is: (f ∘ f)(x) = f(f(x)) = f 2(x)
(f ∘ f ∘ f)(x) = f(f(f(x))) = f 3(x) More generally, for any natural number n ≥ 2,
the nth functional power can be defined inductively by f n = f ∘ f n−1 = f n−1 ∘ f.
Repeated composition of such a function with itself is called iterated function.
By convention, f 0 is defined as the identity map on f 's domain, idX.
If even Y = X and f: X → X admits an inverse function f −1, negative functional powers
f −n are defined for n > 0 as the opposite power of the inverse function: f −n =n.
Note: If f takes its values in a ring, there is a risk of confusion, as f n could also
stand for the n-fold product of f, e.g. f 2(x) = f(x) · f(x). For trigonometric functions,
usually the latter is meant, at least for positive exponents. For example, in trigonometry,
this superscript notation represents standard exponentiation when used with trigonometric
functions: sin2(x) = sin(x) · sin(x). However, for negative exponents, it nevertheless usually
refers to the inverse function, e.g., tan−1 = arctan.
In some cases, when, for a given function f, the equation g ∘ g = f has a unique solution
g, that function can be defined as the functional square root of f, then written as g = f 1/2.
More generally, when gn = f has a unique solution for some natural number n > 0, then f m/n
can be defined as gm. Under additional restrictions, this idea can
be generalized so that the iteration count becomes a continuous parameter; in this case,
such a system is called a flow, specified through solutions of Schröder's equation.
Iterated functions and flows occur naturally in the study of fractals and dynamical systems.
Alternative notations Many mathematicians, particularly in group
theory, omit the composition symbol, writing gf for g ∘ f.
In the mid-20th century, some mathematicians decided that writing "g ∘ f " to mean
"first apply f, then apply g" was too confusing and decided to change notations. They write
"xf " for "f(x)" and "(xf)g" for "g(f(x))". This can be more natural and seem simpler
than writing functions on the left in some areas – in linear algebra, for instance,
when x is a row vector and f and g denote matrices and the composition is by matrix
multiplication. This alternative notation is called postfix notation. The order is important
because matrix multiplication is non-commutative. Successive transformations applying and composing
to the right agrees with the left-to-right reading sequence.
Mathematicians who use postfix notation may write "fg", meaning first apply f and then
apply g, in keeping with the order the symbols occur in postfix notation, thus making the
notation "fg" ambiguous. Computer scientists may write "f ; g" for this, thereby disambiguating
the order of composition. To distinguish the left composition operator from a text semicolon,
in the Z notation the ⨾ character is used for left relation composition. Since all functions
are binary relations, it is correct to use the [fat] semicolon for function composition
as well. Composition operator
Given a function g, the composition operator Cg is defined as that operator which maps
functions to functions as
Composition operators are studied in the field of operator theory.
In programming languages
Function composition appears in one form or another in numerous programming languages.
Multivariate functions Partial composition is possible for multivariate
functions. The function resulting when some argument xi of the function f is replaced
by the function g is called a composition of f and g in some computer engineering contexts,
and is denoted f |xi = g
When g is a simple constant b, composition degenerates into a valuation, whose result
is also known as restriction or co-factor.
In general, the composition of multivariate functions may involve several other functions
as arguments, as in the definition of primitive recursive function. Given f, a n-ary function,
and n m-ary functions g1, ..., gn, the composition of f with g1, ..., gn, is the m-ary function
. This is sometimes called the generalized composite
of f with g1, ..., gn. The partial composition in only one argument mentioned previously
can be instantiated from this more general scheme by setting all argument functions except
one to be suitably chosen projection functions. Note also that g1, ..., gn can be seen as
a single vector/tuple-valued function in this generalized scheme, in which case this is
precisely the standard definition of function composition.
A set of finitary operations on some base set X is called a clone if it contains all
projections and is closed under generalized composition. Note that a clone generally contains
operations of various arities. The notion of commutation also finds an interesting generalization
in the multivariate case; a function f of arity n is said to commute with a function
g of arity m if f is a homomorphism preserving g, and vice-versa i.e.:
. A unary operation always commutes with itself,
but this is not necessarily the case for a binary operation. A binary operation that
commutes with itself is called medial or entropic. Generalizations
Composition can be generalized to arbitrary binary relations. If R ⊆ X × Y and S ⊆ Y
× Z are two binary relations, then their composition S∘R is the relation defined
as {(x, z) ∈ X × Z : ∃y ∈ Y. ∈ R ∧ ∈ S}. Considering a function as a special
case of a binary relation, function composition satisfies the definition for relation composition.
The composition is defined in the same way for partial functions and Cayley's theorem
has its analogue called Wagner-Preston theorem. The category of sets with functions as morphisms
is the prototypical category. The axioms of a category are in fact inspired from the properties
of function composition. The structures given by composition are axiomatized and generalized
in category theory with the concept of morphism as the category-theoretical replacement of
functions. The order inversion in the formula−1 = applies for groups in general and for the
inverse relation; each of these is a dagger category.
See also Combinatory logic
Function composition Functional decomposition
Iterated function Flow
Higher-order function Cobweb plot – a graphical technique for
functional composition Lambda calculus
Functional square root Composition ring, a formal axiomatization
of the composition operation Notes
References ^ a b Daniel J. Velleman. How to Prove It:
A Structured Approach. Cambridge University Press. p. 232. ISBN 978-1-139-45097-3.
^ Nancy Rodgers. Learning to Reason: An Introduction to Logic, Sets, and Relations. John Wiley
& Sons. pp. 359–362. ISBN 978-0-471-37122-9. ^ Christopher Hollings. Mathematics across
the Iron Curtain: A History of the Algebraic Theory of Semigroups. American Mathematical
Society. p. 334. ISBN 978-1-4704-1493-1. ^ Pierre A. Grillet. Semigroups: An Introduction
to the Structure Theory. CRC Press. p. 2. ISBN 978-0-8247-9662-4.
^ Pál Dömösi; Chrystopher L. Nehaniv. Algebraic Theory of Automata Networks: A Introduction.
SIAM. p. 8. ISBN 978-0-89871-569-9. ^ Nathan Carter. Visual Group Theory. MAA.
p. 95. ISBN 978-0-88385-757-1. ^ Olexandr Ganyushkin; Volodymyr Mazorchuk.
Classical Finite Transformation Semigroups: An Introduction. Springer Science & Business
Media. p. 24. ISBN 978-1-84800-281-4. ^ Oleg A. Ivanov. Making Mathematics Come
to Life: A Guide for Teachers and Students. American Mathematical Soc. pp. 217–. ISBN 978-0-8218-4808-1.
^ Jean Gallier. Discrete Mathematics. Springer. p. 118. ISBN 978-1-4419-8047-2.
^ Michael Barr; Charles Wells. Category Theory for Computing Science. p. 6. This is the
updated and free version of book originally published by Prentice Hall in 1990 as ISBN
978-0-13-120486-7. ^ ISO/IEC 13568:2002(E), p. 23
^ Bryant, R.E.. "Logic Minimization Algorithms for VLSI Synthesis". IEEE Transactions on
Computers. C-35 issue=8: 677–691. ^ a b Clifford Bergman. Universal Algebra:
Fundamentals and Selected Topics. CRC Press. pp. 79–80. ISBN 978-1-4398-5129-6.
^ George Tourlakis. Theory of Computation. John Wiley & Sons. p. 100. ISBN 978-1-118-31533-0.
^ a b Clifford Bergman. Universal Algebra: Fundamentals and Selected Topics. CRC Press.
pp. 90–91. ISBN 978-1-4398-5129-6. ^ S. Lipscomb, "Symmetric Inverse Semigroups",
AMS Mathematical Surveys and Monographs, ISBN 0-8218-0627-0, p. xv
^ Peter Hilton; Yel-Chiang Wu. A Course in Modern Algebra. John Wiley & Sons. p. 65.
ISBN 978-0-471-50405-4. External links
Hazewinkel, Michiel, ed., "Composite function", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
"Composition of Functions" by Bruce Atwood, the Wolfram Demonstrations Project, 2007.