Function (mathematics)

In mathematics, a function or mapping is a relation ( relation) between two sets which assigns to each element of one set ( function argument, independent variable, value) exactly one element of the other set ( function value, dependent variable, value). The concept of function is defined differently in the literature, however, is generally based on the idea that functions assign mathematical objects mathematical objects, for example, each real number whose square. The concept of function or mapping takes in modern mathematics a central position; it contains as special cases, among others, parametric curves, scalar and vector fields, transformations, operations, operators, and much more.

• 3.1 spellings
• 3.2 ways of speaking

• 5.1 image and inverse image
• 5.2 injectivity, surjectivity, bijectivity
• 5.3 arity
• 5.4 set of functions

• 6.1 restriction
• 6.2 inverse function
• 6.3 concatenation
• 6.4 link

• 7.1 Algebraic properties
• 7.2 Analytical properties

Conceptual history

The first approaches to an implicit use of the concept of function in tabular form ( shadow length depending on the time of day, depending on the central angle chord lengths, etc ) can already be seen in the ancient world. The first evidence of an explicit definition of the concept of function can be found in Nicholas of Oresme, the dependencies of varying sizes (heat, movement, etc ) graphically by mutually perpendicular distances ( longitudo, latitudo ) represented in the 14th century. At the beginning of the process to develop the concept of function are Descartes and Fermat, who developed the analytical method to the introduction of functions with the help of imported Viète of variables. Functional dependencies should be represented by equations such as. In school mathematics this naive notion of function was maintained until well into the second half of the 20th century. The first definition of the concept of function for this idea comes from Gregory in his 1667 published book Vera Circuli et hyperbolae quadratura. The term function is probably the first time in 1673 in a manuscript of Leibniz, who used in his treatise of 1692 De linea ex lineis numero Infiniti ordinatim ductis the terms " constant", " Variable", " offset" and " abscissa ". In the correspondence between Leibniz and Johann Bernoulli function term is separated from the geometry and transferred to the algebra. In contributions by 1706, 1708 and 1718 Bernoulli, this development dar. 1748 clarified Euler, a student of Johann Bernoulli, in his book Introductio in analysin infinitorum the concept of function on.

In Euler there are two different explanations of the concept of function: On the one hand, each " analytic expression " in x is a function that, on the other hand is y ( x) defined by a freehand curve drawn in the coordinate system. In 1755 he formulated these ideas without the use of the term " analytic expression " to. He also led the 1734 notation f ( x) a. He distinguishes between unique and ambiguous features. Euler thus is also the reverse of the normal parabola, wherein each non-negative real number in both its positive and its negative root is assigned, as a function permitted. For Lagrange only functions are allowed, which are defined by power series, as he defines in his 1797 Théorie des fonctions analytiques. A fruitful discussion about the law of motion of a vibrating string to the d' Alembert in 1747, Euler in 1748, Daniel Bernoulli presented in 1753 various solutions led to the discovery of the set of definitions and a further präzisierten function term in which such a thing as clear assignment already circumscribed by, Fourier in his 1822 book published Théorie de la chaleur analytique. Like Cauchy formulated in 1823 in the Résumé leçons ... sur le calcul infinitésimal.

When the analysis was made in the 19th century with an exact term limit on a new basis, were characteristics that were previously regarded as constitutive for functions, introduced in a Exaktifizierungsprozess as independent terms and detached from the concept of function. Dirichlet, a disciple of Fourier, formulated this new view: "Ideas in the place of bills " and presented his ideas in 1837 dar. Stokes resulted in works from 1848 and 1849 similar views. So proceeded Riemann, a student of Dirichlet, 1851 in foundations for a general theory of functions of a single variable complexen size with the continuity, later followed integrability and differentiability. A summary of this development makes Hankel 1870 in studies of the infinitely oscillierenden and discontinuous functions. Here f (x) at the position x is not distinguished between the function f and the function value.

Weierstrass, Dedekind and others discovered that limits of infinite sequences of "classical" functions can be erratic and can not always be expressed by "closed" formulas, that is, with a finite number of arithmetic operations. This forced a gradual extension of the concept of function.

Irrespective of group theory was established in the 19th century, with which one can systematically investigate how changing algebraic equations under the effect of successive transformations. In applying this theory to geometric problems were synonymous, the terms of movement and figure used with transformation.

When at the beginning of the 20th century the foundations of mathematics were uniformly formulated in the language of set theory, the mathematical concepts and functional imaging were found to be congruent. However, in the parlance of different traditions act on. In calculus, we now speak often of even functions, while speaking in algebra and geometry of pictures. Some mathematicians distinguish strictly between a still image and a function. This meant by a function mapping into the real or complex number field.

Other synonyms for function in more specific contexts include operator in the analysis, operation, logic operation and morphisms in algebra.

Today, some authors do not see the concept of function necessarily limited to quantities on, but let each consisting of ordered pairs class that does not contain different elements with the same left component, considered as function. Functions are expressed quantity theory thus defined as quite distinct relations.

Definition

The basic idea

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

Notation:

For the only, the item associated with the target set to write in general.

Notes:

• The converse is not true: An element of the target set must ( if any) have been not only associated with an element of the definition set.
• Often, instead of the amount initially a source definition given amount. When given as a calculation rule, one obtains the definition size, by excluding from the elements, is not defined for the.

Amount Theoretical Definition

Quantity theory is a function of a special relation:

• Is a subset of the cartesian product of and, i.e., is a ratio.
• Exists for each element of ( at least ) a component in, so that the ordered pair of the element ratio. is thus left total.
• For each element of an element there are at most of, so that the pair is. making it quite clear or functional.

The last two properties can also be summarized as follows:

• For each element of it is exactly one element of, such that the pair element of the relation.

Often, however, you want to make the target set explicitly a part of the function to, for example, statements can do for surjectivity:

Is also known as the graph of the function. The definition of the functional quantity is determined uniquely by its graph and consisting of the first components of all the elements of the graph. If two functions in their graphs coincide, then we also say that they are essentially the same.

However, it can also be affected by the size and definition of a function corresponding to, defined as a triplet, as described above.

Notation

Spellings

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

Ways of speaking

To assign a function value to an argument, there are a number of different speech or detailed notations, which are all more or less equivalent and above all a function of what is to be expressed superficially, on the context, the symbolism used and also from taste of the speaker ( writer ) can be selected. Here are some examples:

Must be distinguished from the talk and writing: "y is a function of x ", mainly in physics, closely related areas of mathematics appears. She is the older and original speech and writing and describes the dependency of one variable on another variable, in contrast, that is described by means of the variables and ( representative ) the assignment of certain elements of sets. The " physical " way of speaking comes from the procedure, first two variable quantities ( the physical reality) symbols, namely the variables and to assign and then determine their dependence. Is, for example, the room temperature and for the time, one is able to determine that the room temperature changes as a function of time, and thus, " the room temperature, a function of time " or representative "y is a function of x. "

Instead of defining quantity and range definition, prototype quantity or plain archetype is said. The elements of are called function arguments or archetypes, casual and x values ​​. The target amount is also called the set of values ​​or range of values ​​, the elements of the hot target values ​​or target elements loosely and y values. Those elements of that actually occur as a picture of an argument, are called function values ​​, image elements or simple images.

Representation

A function can visualize it by drawing its graph in a ( two-dimensional ) coordinate system. The function graph of a function can be defined mathematically as the set of all pairs of elements, for the. The graph of a continuous function to a continuous interval forms a continuous curve (more precisely, the amount of the points of the curve, seen as sub-space of a topological space is connected).

Analogously one can obtain functions and to visualize, by records them in a three-dimensional coordinate system. Is continuous, we obtain a curve ( which may also have corners), the " snakes" by the coordinate system. Is continuous, we obtain a surface as an image, typically in the form of a " mountain ".

Computer programs for representation of functions are called function plotter. Functional programs are also on the features of computer algebra systems ( CAS), matrizenfähigen programming environments such as MATLAB, Scilab, GNU Octave and other systems. The essential skills of a Funktionenplotters are also available on a graphics-capable calculator. There are also web-based offers that only require a current browser.

• Examples of some function graphs

Polynomial function of degree 5

Real part of the complex exponential

Sine function

Spherical harmonic

Bell curve

Basic Properties

Image and preimage

The image of an element of the definition of quantity is simply the function value. The image of a function is the set of images of all elements of the definition of quantity, ie

The image is therefore a function of a subset of the target set, and is referred to image set. The inverse image of an element of the target set is the set of all elements of the definition set whose image is. We write

The inverse image 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 every element of the target set has at most one preimage. Ie follows
• It is surjective if every element of the target set has at least one preimage. That is, a to any there, so that
• It is bijective if it is injective and surjective, ie, if every element of the target set has exactly one preimage.

Arity

A function whose amount is an amount of product is often called double digits. The value of which is obtained with the use of the pair referred to as one.

The same is true for higher arities. A function is usually referred to as three digits. A function whose amount is not a product lot (or in which the inner structure of the set of definitions does not matter) is called a digit. Under a zero- ary function is defined as a function whose amount is the empty product.

Instead of one digit, two digits, three digits are also often says unary, binary, ternary; Arity is therefore also called " arity " (English: arity ) refers.

Set of functions

With or the set of all mappings is denoted by by:

Operations

Restriction

The restriction of a function on a subset of the set of definitions is given by the function whose graph

Is given.

Inverse function

For every bijective function is an inverse function

So that is uniquely determined element applies to the. The inverse function fulfilled for all

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

Concatenation

Two functions and in which the value range of the first function corresponds to the domain of definition of the second function, may be concatenated. The concatenation or sequential execution of these two functions is then a new function defined by

Is given. In this notation, the picture usually is applied first to the right, ie when the function is first applied and then the function. Occasionally, the reverse ranking is used and written in the literature, however.

Link

Is on the target set an internal binary operation given, it can be also functions an internal binary operation defining:

Examples are the pointwise multiplication and addition functions. Next can also define the linkage of a function to an element with the help of an external two -digit shortcut of the form:

Example of this is the pointwise multiplication of a function by a scalar. Analogously, as also define an outer join of the form. Are links of the same type given both on the set of definitions, as well as on the target quantity, ie a function compatible with these links when the pictures behave with respect to a link as well as the inverse images with respect to the other link.

Other properties

Algebraic properties

• A function is idempotent if, that is, for all elements of the definition quantity applies.
• She is an involution if it is so true for all elements of the definition and quantity for at least one of the set of definitions.
• A fixture is an element of the definition set of is considered for the.
• Identity
• Constance

Analytical properties

• Narrowness
• Periodicity
• Monotony
• Symmetry
• Continuity
• Differentiability
• Smoothness
• Holomorphy
• Homogeneity
• Measurability
• Integrability
• Convexity

Special Functions

• Homogeneous linear function (also: proportionality ): generally described by; is a homomorphism with respect to addition
• General linear function (or affine function ): general described by; see also affine
• Quadratic function: general described by (see Quadratic equation )
• Power function
• Polynomial functions; quite rational function: general described by or
• Rational function; fractional-rational function: the quotient of two polynomial functions,
• Root function: consists of fractional rational functions, combined with the basic arithmetic and root expressions
• Exponential
• Logarithm
• Trigonometric functions: sin, cos, tan, cot, sec, csc
• Absolute value function
• Maximum function and minimum function
• Gaussian integer function

Use

A fundamental concept in mathematics represent structures arising from the fact that quantities are taken in conjunction with accompanying illustrations. Such structures form the basis of virtually all mathematical disciplines, once they go combinatorial problems or basic mathematical and philosophical questions on elementary set theory.

Quantities can be structured, for example, by so-called shortcuts. The most important special case is the inner two-digit shortcut, this is an image of the form. Examples of internal double-digit shortcuts are arithmetic operations such as addition or multiplication on sets of numbers. Accordingly, the image of a couple taking a shortcut is usually written in the form.

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

Generalizations

Multi-functions

A multi-function (also called multi-valued function or correspondence ) is a left- total relation. That is, the elements of the set of definitions can be mapped to a plurality of elements of the target set. It also writes. An example of multi-functions are the inverse functions of surjective functions. ( If is surjective, automatically applies: is a multi-function. )

If a lot, then you can represent each multi-function as a function that goes into the power set of.

Partial functions

Wohlzuunterscheiden the concept of function is the notion of partial function, one also speaks of a " not everywhere defined function " or " functional relation". There may be elements of the source set ( values), which have no associated value of the target quantity ( value). Here the mention of the source amount in the above Tripelschreibweise is actually necessary. However, there must be no more of a value than a value there. To distinguish partial functions of functions, the latter is referred to as total or everywhere defined functions.

Functions with values ​​in a real class

Often the values ​​of a function are not in a target amount, but only in a real class, for example, are quantity consequences " functions" with definition of quantity and values ​​in the universal class. To avoid set-theoretic problems arising from the fact, considering only the graph of the corresponding function, more precisely, a function -like graph is a set of pairs, such that no two pairs match in the first entry:

Definition and set of values ​​are actually quantities, but it is not necessary to commit in advance to a target amount.

Symbolism

For functions, there are several symbolic notations, each expressing some special properties of the function. Below are mentioned some important.

The icons can be also, where appropriate, combined.