# Function (mathematics)

a function expresses the dependence of a size of another. Traditionally functions are defined as rule or regulation, which transforms an input (argument, usually x) into an initial value (function value, usually y)(transfers).

Frequently also the terms illustration and operator for functions are used.

In school mathematics one becomes acquainted with first simple functions how:

[itex] y = 2 \ cdot x + 3 [/itex] or [itex] y = x^2 [/itex]

Mathematics defines functionsin the terms of the set theory.

to work on] to definition

a function f also point to everyone Element of a definition quantity of A (a “x-value”) exactly one element of a goal quantity of B (a “y-value”) too. A function has therefore the explicit characteristic:

Each x - Worth from the definition range exactly one y-value is assigned.

Often one can indicate an allocation regulation, oneit calls function equation.

A function a right-on the left and on the right relation is set-theoretical, i.e.:

A function of the quantity of A into the quantity of B is a quantity of f, which have the following characteristics:
• f is a subsetof A × B (cartesian product), thus a quantity of pairs (A, b), whereby A in A and b in B apply.
• to each element A of A there is exactly one element b of B (written f (A)), so that the pair (A, b) is element of f.

Often one in addition, the goal quantity would like to make B explicitly a part of the function, and defined:

A Tripel f = (A, B, R) consisting of two quantities of A and B as well as a relation RA × B are called function from A to B, if applies: to each element there is A of Aexactly one element b of B (written f (A)), so that the pair (A, b) is element of R. A function is thus certain by its graph R and the indication of the quantity B.

Besidesthere is still the term partial function, which particularly in computer science one uses. Here required that a value is assigned to each argument, it is not only required that there is at the most an assigned value. This is not Function in the sense defined here; such hot function total in this context.

## ways of writing and manners of speaking

• [itex] f \ colon A \ tons of B [/itex]
(and/or. f: A -> B in the text mode) instead of [itex] f \ subseteq A \ times B< /math>,
„function f from A to B
• [itex] f \ colon x \ mapsto f (x) [/itex]
(and/or. f: x -> f (x) in the text mode) or y = f (x) instead of [itex] (x, y) \ in f< /math>.
x shown on f by x
x f of x assigned “
y is f of x
„y is the picture of x under the illustration f “.

The definition quantity of A is called also definition range. The elements of A are called function arguments, casually also „x - values “, the elements of B are called casually also „y - values “. Function values are called against it only those elements of B, which actually appear as picture of an argument.

As “worth tightness” or “range of values” becomes somewhat non-uniform

• eitherthe range, thus the quantity {f (x) | x ∈ A} the actually taken values,
• or the goal quantity

designates.

## functions as structures

a large role play functions in mathematics also as helping means,in order to assign several homogeneous size a structure.

Example
around the values 4, 5, to assign to 6 and 4 the structure of a table with two columns and two lines
[itex]
```\ begin {pmatrix}   4 & 5 \ \   6 & 4 \ \ \ end {pmatrix}
```

<to /math>

becomes each positionin the table (line and column represent by the Zahlenpaar) a value assigned, here for example for value 6 in line 2, column 1:
[itex] (2,1) \ mapsto 6< /math>
The function
< math> \ {(1,1), (1,2), (2, 1), (2,2) \} \ tons \ mathbb {R}, \ quad (i, j) \ mapsto a_ {ij}< /math>
is a general representation of such a tablewith values [itex] a_ {11}< /math>, [itex] a_ {12}< /math>, [itex] a_ {21}< /math> and [itex] a_ {22}< /math>.

In this way in mathematics among other things N-tuple, consequences and stencils are defined.

## representation of functions

a function f: RR can one visualize, by drawing its graph into a coordinate system. The function graph of a function f can be mathematically defined as the quantity of all Zahlenpaare (x|y), for y=f (x). The graph of a constant function forms a connected curve.

Computer programs for the representation of functions are called function plotters. Function plotters belong also to the function range of computer algebra systems (CAS), to stencilable programming environments such as MATLAB, Scilab, Octave and other systems. The substantial abilities of a function plotter are available also on an able to graphic representation pocket calculator.

## examples

the normal parabola: [itex] f: \ mathbb {R} \ tons \ mathbb {R}, \; \; x \ mapsto f (x) =x^2< /math>

The successors - function: [itex] s:\mathbb {N} \ tons \ mathbb {N}, \; \; x \ mapsto s (x) =x+1< /math>

## important one of terms

• the picture of an element x of the definition quantity is simply f (x).
• The picture of a function is the quantity of all pictures, thus f (A) = { f (x): x in A }
• the Urbild of an element y of the worth tightness is the quantity of all elements of the definition range,their fig. y is. One writes -1 (y) = {x to f in A : f (x) = y }. One says also fiber of y.
• The Urbild of a subset of M of the goal quantity is thoseMixes all elements of the definition range, whose picture is element of this subset.f -1 (M) = { x in A : f (x) in M }.
• The composition is the linkage of functions by Hintereinanderausführung (f o g) (x) = f (g (x)).
• The inverse function of a bijektiven function assigns the Urbildelement to each element of the worth tightness. (With bijektiven functioning the Urbild of each element has exactly one element.)
• a fixed point is inElement x of the definition range of f, to which f (x) applies = x.

## characteristics of functions

### general characteristics

• a function is injektiv, if each element of the range of values at the most a Urbildhas.
• It is surjektiv, if each element of the goal quantity has at least a Urbild.
• It is bijektiv, if it is injektiv and surjektiv, thus if each element of the goal quantity exactly a Urbild has.
• It is idempotent, if f (f (x))=f (x)to all elements x of the definition range applies.
• It is a Involution, if f (f (x)) = x to all elements x of the definition range applies.
• A two digit function f means commutative, if f (x, y) =f (y, x) for all x and y from the definition quantityapplies.

## functions, the structures consider

functions, those on connections such as z. B. Operations(Addition, etc.) in the definition and to the goal quantity „consideration take “, Morphismen are called. See Homomorphismus, category theory.

## special functions and function types

polynomial function 5. Degree
complex exponential function
sine function
spherical surface function

it gives most different distinguishersand thus also many names for individual function types.

### analytic functions

• algebraic functions a function is algebraic, if it consists only of a linkage of the basic operations of arithmetic and extracting the root.

] is analytic