Contents

1. Overview
2. Etymology
3. Cultural Impact

This article is about the mathematical concept; it is not to be confused with Function composition (computer science) , which, while related, delves into the specifics of its application within computational paradigms.

“Ring operator” redirects here; not to be confused with operator ring or operator assistance . The distinction is, of course, critical for those who value precision.
“∘” redirects here. For the character itself, which has a surprising number of doppelgängers, see Degree symbol § Lookalikes .

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

In the grand, often tedious, theatre of mathematics , the composition operator, denoted by the unassuming little circle

∘

{\displaystyle \circ }

, performs a rather elegant, if somewhat obvious, act. It takes two distinct functions , typically labeled

{\displaystyle f}

and

{\displaystyle g}

, and, with a flourish, synthesizes them into an entirely new function. This new entity, often represented as

h ( x ) := ( f ∘ g ) ( x )

f ( g ( x ) )

{\displaystyle h(x):=(f\circ g)(x)=f(g(x))}

, is the result of applying one function after the other. Specifically, the function f is applied to the output of g, after g has first operated on the input x. It’s a sequential transformation, if you will, where the universe first experiences g, then f.

The expression

( f ∘ g )

{\displaystyle (f\circ g)}

is conventionally pronounced “the composition of f and g.” It’s a phrase that rolls off the tongue with an air of sophisticated simplicity, though its implications can be anything but.

It’s worth noting the concept of “reverse composition,” which, as its name subtly suggests, applies the functions in the inverse sequence. Here, the function

{\displaystyle f}

is applied initially, and its output subsequently serves as the input for function

{\displaystyle g}

. Intuitively, one can visualize reverse composition as a methodical chaining process, where the transformed output of function f seamlessly feeds into the awaiting input of function g. This creates a cascade of operations, each building upon the last.

Function composition isn’t some niche mathematical quirk; it’s a specific manifestation of the more general composition of relations . Consequently, it inherits all the fundamental properties inherent to relation composition, including, most notably, its unwavering associativity. This foundational connection means that many insights gleaned from the broader theory of relations directly translate to the behavior of composite functions, lending a reassuring consistency to the mathematical landscape.

Examples

To illustrate this fundamental concept, let’s consider a few concrete scenarios, both finite and infinite, because, apparently, examples are required for comprehension.

Composition of functions on a finite set : Imagine two functions, f and g, operating on a small, discrete set of elements. Let f = {(1, 1), (2, 3), (3, 1), (4, 2)} and g = {(1, 2), (2, 3), (3, 1), (4, 2)}. Each pair (input, output) defines the mapping. To find g ∘ f, we trace the path:
- For input 1: f(1) = 1. Then g(1) = 2. So, (g ∘ f)(1) = 2.
- For input 2: f(2) = 3. Then g(3) = 1. So, (g ∘ f)(2) = 1.
- For input 3: f(3) = 1. Then g(1) = 2. So, (g ∘ f)(3) = 2.
- For input 4: f(4) = 2. Then g(2) = 3. So, (g ∘ f)(4) = 3. Thus, g ∘ f = {(1, 2), (2, 1), (3, 2), (4, 3)}, precisely as depicted in the accompanying figure. It’s a rather straightforward chain reaction.
Composition of functions on an infinite set : Let’s elevate the complexity slightly. Consider real-valued functions, operating over the vast expanse of the real numbers , denoted by R.
- Suppose f : R → R is defined by f(x) = 2x + 4.
- And g : R → R is defined by g(x) = x³. Now, let’s compose them, because why not?
- ( f ∘ g )( x ) = f ( g ( x )). Here, g(x) acts as the input for f. So, we substitute x³ into f: f ( x³ ) = 2(x³) + 4 = 2x³ + 4.
- ( g ∘ f )( x ) = g ( f ( x )). Conversely, f(x) becomes the input for g. So, we substitute (2x + 4) into g: g (2x + 4) = (2x + 4)³. Notice, if it wasn’t already painfully obvious, that ( f ∘ g )( x ) ≠ ( g ∘ f )( x ). Composition is rarely commutative , a fact that seems to surprise people more often than it should.
A practical, if mundane, application: Imagine an airplane’s altitude at any given time t is described by the function a ( t ). Simultaneously, the air pressure at a specific altitude x is given by the function p ( x ). If you’re keen to know the air pressure around the plane at time t, you’d logically combine these. The resulting function, ( p ∘ a )( t ), precisely represents the pressure surrounding the aircraft at that particular moment. It’s a simple, elegant way to model dependent variables.
Composition of permutations : Functions that merely rearrange elements within a finite set, such as permutations , can also be composed on the same set. This specific instance is known as the composition of permutations, a fundamental operation in combinatorics and group theory, where the order of operations can drastically alter the final arrangement.

Properties

The composition of functions possesses several inherent characteristics, some of which are remarkably useful, others merely inevitable.

The composition of functions is, without fail, associative —a fundamental property directly inherited from its more general ancestor, the composition of relations . This means that if you have three functions, f, g, and h, that are appropriately composable (i.e., their domains and codomains align), then the order in which you group the compositions doesn’t alter the final outcome. Specifically, f ∘ ( g ∘ h ) = ( f ∘ g ) ∘ h . Because the placement of parentheses is ultimately inconsequential to the result, they are typically omitted, simplifying notation for those who can remember such things.

In a rigorously defined context, the composition g ∘ f is only mathematically sound if the codomain of function f precisely matches the domain of function g. However, in a more flexible, perhaps less pedantic, interpretation, it is often considered sufficient for the codomain of f to be an improper subset of the domain of g. This allows for a wider range of composable functions, provided that the output of f can always serve as a valid input for g.

Furthermore, it’s a common and rather convenient practice to implicitly restrict the domain of f to ensure that its outputs strictly fall within the domain of g. For instance, consider the functions f : R → [(−∞,+9]] defined by f ( x ) = 9 − x² and g : [[0,+∞)] → R defined by

g ( x )

{\displaystyle g(x)={\sqrt {x}}}

. While f itself operates on all real numbers R, its output range is limited. The square root function g, however, demands non-negative inputs. To form the composition g ∘ f, we must ensure that 9 − x² ≥ 0, which implies x² ≤ 9, or −3 ≤ x ≤ 3. Therefore, the composition can only be meaningfully defined on the interval [−3,+3]. This tacit restriction is less an imposition and more a necessary condition for the functions to play nicely together.

The image provided offers a clear visual demonstration of this non-commutative nature, showcasing the compositions of two real functions: the absolute value function and a cubic function . When composed in one order, the resulting graph is distinct from when they are composed in the opposite order. This stark difference underscores that the sequence of application profoundly impacts the final transformation.

Functions g and f are said to commute with each other only in the rare instance where g ∘ f = f ∘ g . This commutativity is a special, almost privileged, property, observed only in particular functions and often under very specific conditions. For example, the equation | x | + 3 = | x + 3| holds true only when x ≥ 0. The visual example further reinforces this point, preventing any lingering illusions of universal commutativity.

A composition of one-to-one (injective) functions will invariably yield another one-to-one function. Likewise, the composition of onto (surjective) functions consistently produces an onto function. From these two facts, it logically follows that the composition of two bijections —functions that are both one-to-one and onto—will also result in a bijection. Furthermore, for an invertible composite function, its inverse function exhibits a peculiar, yet predictable, behavior: ( f ∘ g ) −1 = g −1 ∘ f −1 . This means that to reverse a sequence of operations, you must reverse each individual operation and reverse their order. It’s like unpacking a suitcase: the last item you put in is the first one you take out.

For those who dabble in the calculus of change, the derivatives of compositions involving differentiable functions are elegantly determined using the venerable chain rule . This rule provides a systematic method for calculating the rate of change of a composite function. When delving into even more complex rates of change, the higher derivatives of such functions are given by the rather formidable, but highly effective, Faà di Bruno’s formula . This formula offers a generalized expression for the nth derivative of a composite function, a testament to the intricate relationships within calculus.

While the composition of functions might, at a superficial glance, be casually described as a form of multiplication within a function space, it is crucial to recognize that its properties diverge significantly from the more familiar pointwise multiplication of functions. For instance, as previously highlighted, function composition is inherently not commutative , a stark contrast to the commutative nature of pointwise multiplication (where (fg)(x) = f(x)g(x) = g(x)f(x)). This distinction is vital for avoiding conceptual pitfalls and appreciating the unique algebraic structure that composition introduces.

Composition monoids

Main article: Transformation monoid

Consider a scenario where one possesses a collection of functions, say f : X → X and g : X → X, all sharing the same domain and codomain . These are often referred to as transformations because they map elements within a set back onto the same set. With such transformations, one can construct intricate sequences, or “chains,” of these operations composed together, for instance, f ∘ f ∘ g ∘ f. These chains, when considered collectively, exhibit the distinct algebraic structure of a monoid . Such a structure is aptly named a transformation monoid , or, less frequently, a composition monoid. The internal structure of transformation monoids can, in fact, be remarkably complex, revealing surprising patterns and behaviors. A particularly notable example that showcases such complexity is the intriguing de Rham curve , which arises from iterated function systems. The complete collection of all possible functions mapping a set X to itself, f : X → X, forms what is known as the full transformation semigroup or, alternatively, the symmetric semigroup on X. It’s worth a brief pause to acknowledge that one can technically define two distinct semigroups here, depending on whether the semigroup operation is defined as left or right composition of functions.

The image vividly illustrates the non-commutative nature of composition in a geometric context. It shows the composition of a shear mapping (represented in red) and a clockwise rotation by 45° (represented in green). The original object is on the left. The upper sequence demonstrates applying shear first, then rotation. The lower sequence shows applying rotation first, then shear. The distinct final positions of the object clearly demonstrate that changing the order of these geometric transformations alters the outcome.

If the transformations under consideration are all bijective —meaning they are both injective (one-to-one) and surjective (onto), and thus inherently invertible—then the set encompassing all possible combinations of these functions forms a transformation group , also known as a permutation group . In this context, one would say that this group is generated by the initial set of bijective functions.

The grand assembly of all bijective functions f : X → X, commonly termed permutations , forms a group under the operation of function composition. This particular group is famously known as the symmetric group , though it is occasionally referred to as the composition group. A cornerstone result in group theory , Cayley’s theorem , essentially posits that any group can be understood as a subgroup of a symmetric group, up to isomorphism . This theorem is a profound statement about the universality of permutation groups in describing abstract group structures.

Within the realm of the symmetric semigroup (which, recall, comprises all transformations, not just bijective ones), one can still discern a weaker, and notably non-unique, concept of an inverse, often termed a pseudoinverse. This is because the symmetric semigroup itself is a regular semigroup , a property that guarantees the existence of such generalized inverses, albeit without the strict uniqueness found in groups.

Functional powers

Main article: Iterated function

When the codomain of a function is a subset of its domain —that is, if Y ⊆ X for a function

f : X → Y

{\displaystyle f:X\to Y}

—then the function is capable of composing with itself. This repetitive self-composition is sometimes concisely denoted as

{\displaystyle f^{2}}

, though this notation carries a potential for ambiguity that we’ll address shortly.

To be explicit:

The composition of f with itself, performed once, is:

( f ∘ f ) ( x )

f ( f ( x ) )

( x )

{\displaystyle (f\circ f)(x)=f(f(x))=f^{2}(x)}

Composing f three times yields:

( f ∘ f ∘ f ) ( x )

f ( f ( f ( x ) ) )

( x )

{\displaystyle (f\circ f\circ f)(x)=f(f(f(x)))=f^{3}(x)}

And, extending this pattern, four compositions:

( f ∘ f ∘ f ∘ f ) ( x )

f ( f ( f ( f ( x ) ) ) )

( x )

{\displaystyle (f\circ f\circ f\circ f)(x)=f(f(f(f(x))))=f^{4}(x)}

More generally, for any natural number n ≥ 2, the nth functional power can be defined through a process of induction: f n = f ∘ f n −1 = f n −1 ∘ f . This notation, for those keeping score, was introduced by the likes of Hans Heinrich Bürmann and John Frederick William Herschel . The repeated composition of a function with itself is a field of study in its own right, known as function iteration .

By convention, f 0 is defined as the identity map on f ’s domain , denoted id X . This means f 0 (x) = x, effectively doing nothing, which is precisely what one would expect from a “zero power” of an operation.
Should Y = X and the function f : X → X possess an inverse function , denoted f −1 , then negative functional powers, f − n , are defined for n > 0. These are understood as the negated power of the inverse function: f − n = ( f −1 ) n . This elegantly extends the concept of iteration to “undoing” the function’s action multiple times.

A crucial point of clarification: If f takes its values within a ring (which is particularly relevant for real or complex-valued functions), there exists a genuine risk of confusion. The notation f n could, in this context, also signify the n-fold product of f, meaning f 2 ( x ) = f ( x ) · f ( x ). For trigonometric functions , it is almost universally the latter meaning that is intended, at least when dealing with positive exponents. For example, in trigonometry , sin²(x) explicitly means sin(x) · sin(x), representing standard exponentiation of the function’s value.

However, the waters become muddied with negative exponents. For these, especially −1, the notation almost always reverts to referring to the inverse function . For instance, tan −1 consistently denotes the arctangent function , not 1/tan(x). Context, as always, is paramount, and a source of perpetual low-grade annoyance for those who prefer absolute clarity.

In certain specific cases, where for a given function f, the equation g ∘ g = f possesses a unique solution g, that particular function can be formally defined as the functional square root of f. It is then elegantly written as g = f 1/2 . This extends the concept of roots from numbers to functions, seeking a function that, when composed with itself, yields the original.

More generally, if the equation g n = f has a unique solution for some natural number n > 0, then the fractional functional power f m / n can be defined as g m . This allows for a rational “iteration count.”

Under a set of additional, often stringent, restrictions, this idea can be further generalized. The iteration count can transcend discrete integers and become a continuous parameter. When this occurs, such a system is termed a flow , and its behavior is precisely specified through the solutions of Schröder’s equation . Iterated functions and flows are not mere theoretical constructs; they emerge quite naturally and are indispensable in the study of intricate fractals and the often chaotic realm of dynamical systems .

To diligently avoid any lingering ambiguity, some mathematicians, particularly those who value explicit clarity (and who can blame them?), choose to employ the composition symbol ∘ to unequivocally denote the compositional meaning. They might write f ∘ n ( x ) for the n-th iterate of the function f ( x ), as exemplified by f ∘3 ( x ) meaning f ( f ( f ( x ))). For this very same purpose, Benjamin Peirce utilized the notation f [ n ] ( x ), while Alfred Pringsheim and Jules Molk proposed n f ( x ) as an alternative. These attempts at disambiguation highlight the ongoing struggle for clarity in mathematical notation, a struggle that, frankly, should have been settled long ago.

Alternative notations

Many mathematicians, particularly those immersed in group theory , often forgo the explicit composition symbol, choosing instead to write gf as a shorthand for g ∘ f . This compact notation, while efficient, relies on an unspoken understanding of the operation involved.

During the mid-20th century, a faction of mathematicians embraced postfix notation . In this system, one would write xf to denote f ( x ), and ( xf ) g to represent g ( f ( x )). This approach can feel more intuitive than the traditional prefix notation in various contexts. A prime example is in linear algebra , where x might be a row vector , and f and g represent matrices . The composition then corresponds to matrix multiplication . The order here is paramount, given that function composition is not inherently commutative. With postfix notation, successive transformations apply and compose from left-to-right, aligning rather neatly with the natural left-to-right reading sequence, thus offering a certain cognitive ease.

Mathematicians who adopt postfix notation might then write " fg “, intending to first apply f and then apply g. This, regrettably, introduces an element of ambiguity, as the notation " fg " could be interpreted in multiple ways depending on the conventions of the field. To resolve this potential confusion, computer scientists, ever practical, frequently employ " f ; g " to explicitly define the order of composition. Furthermore, to differentiate the left composition operator from a mere textual semicolon, the Z notation utilizes the dedicated ⨾ character for left relation composition . Given that all functions can be understood as specialized binary relations , it is perfectly valid to employ this “fat” semicolon for function composition as well. More intricate details regarding this notation can, naturally, be found in the article dedicated to the composition of relations .

Composition operator

Main article: Composition operator

Given an arbitrary function g , one can define a specialized entity known as the composition operator , denoted C g . This operator is, in essence, a function that maps other functions to new functions. Specifically, it transforms any function f into the composite function f ∘ g, as expressed by:

f

f ∘ g .

{\displaystyle C_{g}f=f\circ g.}

These composition operators are not mere curiosities; they are deeply investigated within the sophisticated realm of operator theory , where their properties and behaviors reveal much about the structure of function spaces and transformations.

In programming languages

Main article: Function composition (computer science)

The concept of function composition, a cornerstone of mathematical operations, manifests in various forms and syntaxes across countless programming languages . Its utility in structuring code, promoting modularity, and enabling complex data transformations is widely recognized. From functional programming paradigms, where it is a central tenet, to imperative and object-oriented languages that provide mechanisms for chaining operations, the underlying principle of feeding the output of one function as the input to another remains a powerful abstraction tool for developers seeking to build robust and maintainable software systems.

Multivariate functions

The elegance of composition isn’t confined to functions of a single variable. Partial composition is entirely possible, and often necessary, for multivariate functions . When a specific argument, say x i, of a function f is replaced by another function g, the resulting function is referred to as a composition of f and g in certain computer engineering contexts. This operation is denoted as:

= g

= f (

, … ,

i − 1

, g (

, … ,

) ,

i + 1

, … ,

) .

{\displaystyle f|{x{i}=g}=f(x_{1},\ldots ,x_{i-1},g(x_{1},x_{2},\ldots ,x_{n}),x_{i+1},\ldots ,x_{n}).}

This allows for targeted substitution, embedding one computational process directly into another’s argument slot.

When the function g is a mere constant, say b, this form of composition gracefully degenerates into a (partial) valuation. The outcome of such a valuation is also known as a restriction or co-factor, effectively fixing one of the input variables:

= b

= f (

, … ,

i − 1

, b ,

x i + 1

, … ,

) .

{\displaystyle f|{x{i}=b}=f(x_{1},\ldots ,x_{i-1},b,x_{i+1},\ldots ,x_{n}).}

In a broader sense, the composition of multivariate functions can involve several other functions serving as arguments simultaneously, a concept crucial in the definition of a primitive recursive function . Given an n-ary function f (a function taking n arguments) and n distinct m-ary functions g 1 , …, g n (each taking m arguments), the generalized composition of f with g 1 , …, g n produces a new m-ary function:

h (

, … ,

)

f (

(

, … ,

) , … ,

(

, … ,

) ) .

{\displaystyle h(x_{1},\ldots ,x_{m})=f(g_{1}(x_{1},\ldots ,x_{m}),\ldots ,g_{n}(x_{1},\ldots ,x_{m})).}

This is sometimes referred to as the generalized composite or superposition of f with g 1 , …, g n . The partial composition in only one argument, which we discussed previously, can be understood as a specific instance of this more general scheme. This is achieved by setting all argument functions, save one, to be appropriately chosen projection functions (functions that simply return one of their inputs). In this generalized framework, the collection g 1 , …, g n can be conceptually viewed as a single vector- or tuple -valued function, making this precisely the standard definition of function composition, but extended to multiple dimensions.

A collection of finitary operations defined on some base set X is designated a clone if it encompasses all projection functions and maintains closure under this generalized composition. Clones are sophisticated algebraic structures that typically contain operations of varying arities (number of arguments). The notion of commutation, so vital for unary functions, also finds an intriguing generalization within the multivariate domain. A function f of arity n is said to commute with a function g of arity m if f acts as a homomorphism preserving g, and vice versa. This condition is formally expressed as:

f ( g (

, … ,

1 m

) , … , g (

n 1

, … , g (

n m

) )

g ( f (

, … ,

n 1

) , … , f (

1 m

, … ,

n m

) ) .

{\displaystyle f(g(a_{11},\ldots ,a_{1m}),\ldots ,g(a_{n1},\ldots ,a_{nm}))=g(f(a_{11},\ldots ,a_{n1}),\ldots ,f(a_{1m},\ldots ,a_{nm})).}

While a unary operation trivially commutes with itself, this is not a universal truth for binary or higher-arity operations. A binary (or higher arity) operation that possesses the property of commuting with itself is given the rather descriptive title of medial or entropic .

Generalizations

The concept of composition extends far beyond the confines of functions, finding a natural generalization in the realm of arbitrary binary relations . If R is a binary relation from set X to set Y (R ⊆ X × Y), and S is a binary relation from set Y to set Z (S ⊆ Y × Z), then their composition, R ∘ S, is defined as:

R ∘ S

{ ( x , z ) ∈ X × Z : ( ∃ y ∈ Y ) ( ( x , y ) ∈ R

∧

( y , z ) ∈ S ) }

{\displaystyle R\circ S={(x,z)\in X\times Z:(\exists y\in Y)((x,y)\in R,\land ,(y,z)\in S)}}

This definition essentially states that an element x is related to z by the composite relation R ∘ S if there exists at least one intermediate element y in Y such that x is related to y by R, and y is related to z by S. This is a direct extension of the “chaining” idea from functions.

Given that a function is, fundamentally, a specialized instance of a binary relation (specifically, a functional relation where each input maps to exactly one output), it’s no surprise that function composition fits perfectly within the broader definition for relation composition. The same small circle, R ∘ S, has been consistently employed for the infix notation of composition of relations , just as it is for functions. However, when specifically representing the composition of functions,

( g ∘ f ) ( x )

g ( f ( x ) )

{\displaystyle (g\circ f)(x)\ =\ g(f(x))}

, the textual sequence of f and g is reversed relative to the composition symbol. This is a deliberate choice, intended to clearly illustrate the distinct operational sequence: f acts first, then g.

The definition of composition extends seamlessly to partial functions , which are functions that may not be defined for every element in their domain . Furthermore, Cayley’s theorem , which asserts that every group is isomorphic to a subgroup of a symmetric group, finds its relational analogue in the form of the Wagner–Preston theorem , extending the profound connection between abstract algebraic structures and concrete transformations.

The category of sets , with functions serving as its morphisms , stands as the quintessential, prototypical example of a category . Indeed, the very axioms that define a category are, in fact, directly inspired by the fundamental properties (and, naturally, the definition itself) of function composition. This deep connection underscores the foundational role of composition in abstract algebra and higher mathematics. The structures elucidated by composition are meticulously axiomatized and generalized within category theory , where the concept of a morphism serves as the category-theoretical equivalent of functions. It’s noteworthy that the reversed order of composition observed in the formula for an inverse, ( f ∘ g ) −1 = ( g −1 ∘ f −1 ), also applies to the composition of relations when utilizing converse relations , and consequently, holds true in group theory . These intricate structures, where operations can be reversed and composed, form the basis of dagger categories .

The conventional “foundation” for mathematics typically begins with the bedrock concepts of sets and their elements . However, it is entirely possible, and indeed fruitful, to embark on a different foundational journey—one that axiomatizes not the elements within sets, but rather the functions between sets. This alternative perspective can be elegantly articulated using the language of categories and universal constructions.

As Saunders Mac Lane so eloquently put it in his seminal work, Mathematics: Form and Function : " . . . the membership relation for sets can often be replaced by the composition operation for functions. This leads to an alternative foundation for Mathematics upon categories – specifically, on the category of all functions. Now much of Mathematics is dynamic, in that it deals with morphisms of an object into another object of the same kind. Such morphisms ( like functions ) form categories, and so the approach via categories fits well with the objective of organizing and understanding Mathematics. That, in truth, should be the goal of a proper philosophy of Mathematics.” This quote highlights a profound shift in perspective, elevating composition to a foundational role, and, frankly, it’s about time someone articulated it so clearly.

Typography

The symbol employed for composition, ∘, is encoded in Unicode as U+2218 ∘ RING OPERATOR ( ⋅ ∘, ∘). For those with a keen eye for detail, the Degree symbol article provides a helpful comparison with other Unicode characters that bear a similar visual resemblance, ensuring no undue confusion arises. In the esteemed typesetting system TeX , this symbol is rendered using the command \circ.

Notes

^ The strict sense, where the codomain of f must precisely equal the domain of g, is rigorously upheld, for instance, in category theory . Here, any instance of a subset relation is not merely assumed but is explicitly modeled by an inclusion function , ensuring formal precision. It’s a level of exactitude that, frankly, some find excessive.
^ It is imperative that Alfred Pringsheim ’s and Jules Molk ’s (1907) notation n f ( x ) for denoting function compositions is not conflated with Rudolf von Bitter Rucker ’s (1982) notation n x . The latter, introduced by Hans Maurer (1901) and Reuben Louis Goodstein (1947), specifically refers to tetration . Similarly, it must also not be confused with David Patterson Ellerman ’s (1995) n x pre-superscript notation, which is used to represent roots . The subtleties of notation are, it seems, a constant source of potential misinterpretation.

Function Composition

h ( x ) := ( f ∘ g ) ( x )

Examples

Properties

g ( x )

Composition monoids

Functional powers

( f ∘ f ) ( x )

f ( f ( x ) )

( f ∘ f ∘ f ) ( x )

f ( f ( f ( x ) ) )

( f ∘ f ∘ f ∘ f ) ( x )

f ( f ( f ( f ( x ) ) ) )

Alternative notations

Composition operator

f

In programming languages

Multivariate functions

)

) )

Generalizations

R ∘ S

( g ∘ f ) ( x )

Typography

See also

Notes