Function (math)

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In mathematics , a function ( Latin functio ) or mapping is a relationship ( relation ) between two sets , which gives each element of one set (function argument, independent variable, value) exactly one element of the other set (function value, dependent variable, value ) assigns. The term function is defined differently in the literature, but one generally starts from the idea that functions assign mathematical objects to mathematical objects, for example every real number its square. The concept of function or mapping occupies a central position in modern mathematics; As special cases it contains among other things parametric curves , scalar and vector fields , transformations , operations , operators and much more.

Concept history

The first approaches to an implicit use of the functional term in tabular form (shadow length depending on the time of day, chord length depending on the central angle, etc.) can already be seen in antiquity. The first evidence of an explicit definition of the concept of function can be found in Nikolaus von Oresme , who in the 14th century graphically represented the dependencies of changing quantities (heat, movement, etc.) by means of perpendicular lines (longitudo, latitudo). At the beginning of the process of developing the concept of function are Descartes and Fermat , who developed the analytical method of introducing functions with the help of the variables introduced by Vieta . Functional dependencies should be represented by equations such as . In school mathematics, this naive function concept was retained well into the second half of the 20th century. The first paraphrase of the concept of function based on this idea comes from Gregory in his book Vera circuli et hyperbolae quadratura , published in 1667 . The term function probably first appeared in 1673 in a manuscript by Leibniz , who also used the terms “constant”, “variable”, “ordinate” and “abscissa” in his 1692 treatise De linea ex lineis numero infinitis ordinatim ductis . In the correspondence between Leibniz and Johann I Bernoulli , the concept of function is detached from geometry and transferred to algebra. Bernoulli describes this development in contributions from 1706, 1708 and 1718. In 1748 Leonhard Euler , a pupil of Johann Bernoulli, refined the concept of function in his book Introductio in analysin infinitorum .

Euler has two different explanations of the concept of function: On the one hand, each “analytical expression” is represented in a function, and on the other hand, it is defined in the coordinate system by a freehand drawn curve. In 1755 he reformulated these ideas without using the term "analytical expression". In addition, he introduced the spelling as early as 1734 . He differentiates between unique and ambiguous functions. With Euler, the inversion of the normal parabola , in which each non-negative real number is assigned both its positive and its negative root, is allowed as a function. For Lagrange , only functions are allowed that are defined by power series, as he stated in his Théorie des fonctions analytiques in 1797 . A fruitful discussion about the law of motion of a vibrating string, for which d'Alembert in 1747, Euler 1748 and Daniel Bernoulli presented different solutions in 1753, led to the discovery of the set of definitions and a more precise concept of function, in which something like a clear assignment is already described Fourier in his book Théorie analytique de la chaleur , published in 1822 . Cauchy formulated something similar in 1823 in Résumé des leçons… sur le calcul infinitésimal.

When analysis was put on a new basis with an exact limit value concept in the 19th century , properties that were previously understood as constituting functions were introduced as independent concepts in an exactification process and separated from the function concept. Dirichlet , a student of Fourier, formulated this new view: "Ideas in place of bills" and presented his ideas in 1837. Stokes carried out similar views in works in 1848 and 1849. This is how Riemann , a pupil of Dirichlet, proceeded in 1851 in the foundations of a general theory of the functions of a variable complex quantity with continuity, later integrability and differentiability followed. Hankel summarized this development in 1870 in his investigations into the infinitely often oscillating and discontinuous functions. Here, too, no distinction is made between the function and the function value at the point .

Weierstraß , Dedekind and others discovered that limit values ​​of infinite sequences of "classic" functions can be erratic and not always through "closed" formulas, ie. H. with a finite number of arithmetic operations. This forced a gradual expansion of the concept of function.

Independent of this, group theory was founded in the 19th century , with which one can systematically examine how algebraic equations change under the effect of successive transformations. When applying this theory to geometric problems, the terms movement and mapping were used synonymously with transformation .

When the basics of mathematics were uniformly formulated in the language of set theory at the beginning of the 20th century , the mathematical terms function and mapping turned out to be congruent. However, the different traditions continue to have an effect on language usage. In analysis today one often speaks of functions, while in algebra and geometry one speaks of mappings. Even today, some mathematicians make a strict distinction between a mapping and a function. These understand a function as a mapping into the real or complex number field ( or ) or also powers of it ( or ), on the other hand it is common in Boolean algebra to speak of Boolean functions .

Other synonyms for function in more specific contexts include operators in analysis, operation, linkage and (somewhat generalized) morphism in algebra.

Today, some authors consider the concept of function (as well as the relational concept) is not necessarily as to amounts limited to, but let each of ordered pairs existing class that contains no different elements with the same left component, are considered functional. Expressed in set theory, functions are defined as right-unambiguous relations .


Basic idea

A function assigns exactly one element of a target set to each element of a definition set .


, or equivalent:  

In general, one writes for the element of the target set assigned to the element .


  • The reverse does not apply: An element of the target set can be assigned to exactly one, several, but also no element of the definition set.
    Example: The amount function assigns the numbers +1 and −1 of the definition set to the number +1 of the target set. The number +1 of the target set is assigned two numbers of the definition set, the number −1 is not assigned any number of the definition set.
  • Often, instead of the definition set, a source set is given first. If the calculation rule is given, the definition set is obtained by excluding those elements from for which is not defined. See also section “ Partial Functions ”.

Set theoretical definition

In set theory, a function is a special relation :

A quantity- to-quantity function is a quantity that has the following properties:
  • is a subset of the Cartesian product of and , i. H. is a relation between and .
  • For each element from there is at least one element in , so that the ordered pair is an element of the relation . is left total.
  • For each element of there are at most one element of , so that the pair in located. is therefore legally unambiguous or functional.

The last two properties can also be summarized as follows:

  • For every element of there is exactly one element of , so that the pair is an element of the relation .

Often, however, one would also like to make the target set explicitly part of the function, for example to be able to make statements about surjectivity (as a property of the function itself):

A pair , consisting of a set and a set of pairs with another set, is called a function of the set after , if the following applies: For every element of there is exactly one element of (written ), so that the pair is an element of .

is then also called the graph of the function . The definition set of the function is clearly determined by its graph and consists of the first components of all elements of the graph. If two functions match in their graphs, one also says that they are essentially the same. In particular, each function is essentially the same as the surjective function with the image set .

It is often advisable to add the definition set and define a function as a triple . This definition then agrees with the corresponding detailed definition for relations, so that multifunction and partial functions are also recorded in the same way.



An assignment can be described in one of the following forms, among others:

Function equation with definition set
Clear assignment rule (English: maplet ) with definition set
Clear assignment rule with definition and target quantity
, or equivalent:  
Family spelling (called the index set for the definition set)
Table of values ​​(for finite but also countably infinite sets of definitions)
1 2 3 4th 5 6th 7th ...
1 4th 9 16 25th 36 49 ...
As a relation, in particular, a subset shown as enumerated or described
As a result of links and operations (for example composition , differentiation, formation of the inverse function, ...) that are applied to other functions

Ways of speaking

For the assignment of a function value to an argument, there are a number of different ways of speaking or detailed spelling, all of which are more or less equivalent and, above all, depending on what is to be expressed ostensibly, on the respective context, the symbolism used and also on the The speaker's (writer's) taste can be chosen. Here are some examples:

is mapped to by
from is clearly assigned (primarily if the symbol is in the symbol)
equal to (especially if there is an equal sign in the symbolism)
is the picture from below the picture

This is to be distinguished from the way of speaking and writing: " is a function of ", which appears especially in areas of mathematics that are very closely related to physics. It is the older and original way of speaking and writing and describes the dependency of one variable on another variable , in contrast to the fact that with the help of the variables and (representative) the assignment of certain elements of sets is described. The “physical” way of speaking comes from the procedure of first assigning two variable quantities (of physical reality) symbols, namely the variables and , and then determining their dependency. If, for example, stands for the room temperature and for the time, you will be able to determine that the room temperature changes as a function of time and thus "the room temperature is a function of time" or, as a representative, "is a function of ."

Instead of a set of definitions , domain of definition, set of archetypes or simply archetype are also used . The elements of are called function arguments, function points or archetypes, casually also values. The target set is also called the set of values or the range of values , the elements of hot target values or target elements, also casually -values . Those elements of that actually appear as an image of an argument are called function values, image elements or simply images.


A function can be visualized by drawing its graph in a (two-dimensional) coordinate system . The function graph of a function can be mathematically defined as the set of all element pairs for which is. The graph of a continuous function on a connected interval forms a connected curve (more precisely: the set of points on the curve, understood as a subspace of topological space, is connected).

Similarly, functions , and , can be visualized by drawing them in a three-dimensional coordinate system. If it is continuous, the result is a curve (which can also have corners) that "snakes" through the coordinate system. If continuous, an area emerges as an image, typically in the form of a "mountain landscape".

Computer programs for representing functions are called function plotters . Function programs are also part of the functional scope of computer algebra systems (CAS), matrix-capable programming environments such as MATLAB , Scilab , GNU Octave and other systems. The essential capabilities of a function plotter are also available on a graphical pocket calculator . There are also web-based offers that only require an up-to-date browser.

Basic characteristics

Image and archetype

The image of an element of the definition set is simply the function value . The picture of a function is the set of pictures of all elements of the definition set , that is


The image of a function is therefore a subset of the target set and is called the image set. More generally, is a subset of , then is

the picture from under the function .

The archetype of an element of the target set is the set of all elements of the definition set whose image is. It is


( is generally not a unique function, but a multifunction , for notation see there, as well as under relation #Relations and functions and correspondence (mathematics) ).

Often these fibers are simply referred to as, which in the case of (unambiguously) reversible functions denotes x on the one hand and { x } on the other .

The archetype of a subset of the target set is the set of all elements of the definition set whose image is an element of this subset:


Injectivity, surjectivity, bijectivity

  • A function is injective if each element of the target set has at most one archetype. That is, it follows from
  • It is surjective if every element of the target set has at least one archetype. That is, for anything there is such a thing
  • It is bijective when it is injective and surjective, i.e. when every element of the target set has exactly one archetype.


A function whose definition set is a product set is often called a two-digit number . The value of obtained by applying to the pair is denoted by .

The same applies to higher arity. A function is usually referred to as a three-digit number. A function whose definition set is not a product set (or in which the internal structure of the definition set is irrelevant) is called a one- digit function. A zero-digit function is a function whose definition set is the empty product for any function value. Therefore, zero-digit functions can be understood as constants , which is used in algebraic structures (as well as in heterogeneous algebras ).

Instead of zero-digit, single-digit, two-digit, three-digit one also often says unary, binary, ternary; Arity is therefore also referred to as "arity".

Set of functions

The set of all images from to is denoted by or :

The following applies to the thickness :



The restriction of a function to a subset of the definition set is the function whose graph is by

given is.

Inverse function

For every bijective function there is an inverse function


so that is the uniquely determined element for which applies. The inverse function thus fulfills for everyone


Bijective functions are therefore also referred to as clearly reversible functions.


Two functions and , for which the value range of the first function corresponds to the definition range of the second function (or is contained as a subset), can be concatenated. The chaining or sequential execution of these two functions is then a new function that is carried out by

given is. In this notation, the first figure used is usually on the right, that is, the function is used first and then the function . Occasionally, however, the reverse order is used and written in the literature .


A two-digit link is a figure that maps all pairs of arguments and the final result .

If there is an inner two-digit link on the target set, an inner two-digit link can also be defined for functions :


Examples of this are the pointwise addition and multiplication of functions . With the help of an outer two-digit link of the form , the link between a function and an element can also be defined:

An example of this is the point-by-point multiplication of a function with a scalar . In this way, an external link between the form can also be defined in the same way. If there are links of the same kind on the definition set as well as on the target set, then a function is said to be compatible with these links if the images behave in the same way with respect to one link as the archetypes with respect to the other link.

Other properties

Algebraic properties

  • A function is idempotent if is, i.e. H. applies to all elements of the definition set.
  • It is an involution if is, i.e. holds for all elements of the definition set and is for at least one of the definition set .
  • A fixed point is an element of the definition set of for which applies.
  • identity
  • Constancy

Analytical properties

Special functions


A fundamental concept in mathematics is represented by structures that arise from the fact that sets are seen in connection with associated images. Such structures form the basis of practically all mathematical disciplines as soon as they go beyond elementary set theory, combinatorial problems or fundamental mathematical-philosophical questions.

Quantities can, for example, be structured using so-called links . The most important special case is the inner two-digit link , which is a mapping of the shape . Examples of inner two-digit links are arithmetic operations, such as addition or multiplication on sets of numbers. Accordingly, the image of a couple under a link is usually written in the form .

Other important examples of such structures are algebraic , geometric and topological structures, such as scalar products , norms and metrics .



A multifunction (also called multi-valued function or correspondence) is a left-total relation. This means that the elements of the definition set can be mapped to several elements of the target set . One also writes .

If a lot, then you can run any multi-function also as a function representing, in the power set of works: . 

In this case , a multi-valued function represents a transition relation and is the associated transition function.

The concatenation of multifunctional functions can be defined in the same way as for (unambiguous) functions; in set theory this is equivalent to a concatenation of two two-digit relations .

Inversions of functions as multifunctions

An example of multi-functions are the inverse functions (inversions) of non-injective functions. If is surjective, it automatically applies: is a multifunction. The representation of the inverse function in the power set of yields with the fibers of ( see above ).

The concatenation of a function with its (generally not unambiguous) inversion in the form is an equivalence relation, that of induced equivalence relation . Two elements from the domain are equivalent if and only if they have the same function value.

Partial functions

The partial function and its subset, the function, as special relations

The concept of partial function is to be distinguished from the concept of function , one also speaks of a “function not defined everywhere” or “ functional relation ”. Here there may be elements of the source set ( values) to which no value of the target set ( value) is assigned. In this case, it is actually necessary to name the source quantity in the above triple notation. However, there can not be more than one value for a value . In order to distinguish partial functions from functions, the latter are also referred to as total or universally defined functions.

The set of partial maps from to is the union of the total mappings of subsets from to :

If the sets are finite, then applies to their cardinal numbers


Finally, every partial mapping to D can be reversibly and uniquely extended to a total mapping by fixing any fixed function value that is not contained in; and this operation provides a bijection to represent.

Every partial function is essentially the same as the (total) function with the pre-image set .

Functions with values ​​in a real class

Often the values ​​of a function are not in a target set, but only in a real class , for example set sequences are “functions” with a definition set and values ​​in the universal class . In order to avoid the set-theoretical problems that result from this, one only considers the graph of the corresponding function, more precisely: A function-like graph is a set of pairs , so that no two pairs in the first entry match:

The sets of definitions and values ​​are actually sets, but it is not necessary to commit to a target set in advance as long as the functions are essentially the same.

In the case of partial functions, the same applies to the target and source areas. Both can be real classes individually or together; Set theoretic problems do not arise as long as the graph remains a set.


There are a number of symbolic notations for functions, each expressing some special properties of the function. Some important ones are mentioned below.

symbol Explanation
Function from to

Function on maps; instead of a term or similar. stand

Function on maps; instead of a formula or something similar. stand (set theoretical notation)
Function on maps that the element by element mapping with description of function symbols (instead are often things like u. Ä.) And formula o. Ä. (in place of ) for calculating the image
The most detailed notation, the all involved quantities and the element-wise assignment with description of the function symbols and the formula or similar. to calculate the image
surjective function ( surjection ) from to
injective function ( injection ) from to


bijective function ( bijection ) from to
Inclusion mapping , natural inclusion, natural embedding of in ( is a subset of , and the function maps each element of to itself.)

Identity, identical mapping to A or from to ( and the function maps each element to itself.)

Isomorphism from to

partial function (see above) from to
multi-valued function, multifunction , correspondence (see above) from to

(or ...)
Set of functions (or partial functions), ... from to 

The symbols can also be combined with one another where appropriate.


  • Heinz-Dieter Ebbinghaus : Introduction to set theory. 4th edition. Spectrum, Academic Publishing House, Heidelberg a. a. 2003, ISBN 3-8274-1411-3 .
  • Paul R. Halmos : Naive set theory (= modern mathematics in elementary representation. Vol. 6). Translated by Manfred Armbrust and Fritz Ostermann. 5th edition. Vandenhoeck & Ruprecht, Göttingen 1994, ISBN 3-525-40527-8 .
  • Arnold Oberschelp : General set theory. BI-Wissenschafts-Verlag, Mannheim u. a. 1994, ISBN 3-411-17271-1 .
  • Adolf P. Youschkevitch: The Concept of Function up to the Middle of the 19th Century. In: Archive of the History of Exakt Sciences. 16 Springer Verlag, Berlin 1976.

Web links

Wikibooks: Math for Non-Freaks: Illustration, Function  - Learning and Teaching Materials
Wiktionary: Function  - explanations of meanings, word origins, synonyms, translations
Commons : Functions  - Collection of pictures, videos and audio files

References and comments

  1. M. Kronfellner: Historical aspects in mathematics lessons . Verlag Hölder-Pichler-Tempsky, Vienna 1998, p. 67.
  2. Adolf P. Youschkevitch: The Concept of Function up to the Middle of the 19th Century. In: Archive of the History of Exakt Sciences. 16, Springer Verlag, Berlin 1976, p. 52.
  3. D. Rüthing: Some historical stations on the concept of function. In: Mathematics Lessons. Issue 6/1986, Friedrich Verlag Velber, pp. 5-6.
  4. H.-J. Vollrath: Algebra in Secondary School. BI Wissenschaftsverlag, Mannheim 1994, p. 118.
  5. Rüthing, pp. 6-12.
  6. Arnold Oberschelp: General set theory. 1994.
  7. called class function, see Claudius Röhl: Das Wahllaxiom , diploma thesis Univ. Leipzig, Faculty of Mathematics, October 6, 2016, page 18
  8. ^ Paul R. Halmos: Naive set theory . 1994, Chapter 8, pp. 43 .
  9. less often equivalent based on the set notation
  10. a b c partly also notated without the square brackets
  11. or according to the simplified function definition with function = graph. Alternative notations:
    • or for the correspondence to the multifunction , in the case ( transition function ) too
    • or for the power set of
  12. a b H. König: Design and structural theory of controls for production facilities (=  ISW research and practice . Volume 13 ). Springer-Verlag, Berlin / Heidelberg 1976, ISBN 3-540-07669-7 , pp. 15-17 , doi : 10.1007 / 978-3-642-81027-5_1 . Here: page 21f
  13. as always for two-digit relations; we understand the function as a two-digit relation, especially its inversion
  14. ^ Nicolas Bourbaki : Eléments de mathématiques. Theory of the ensemble. II.
  15. The notation is sometimes used differently for (any) relations.