Τρίτη 19 Αυγούστου 2014

FUNCTION(MATHEMATICS)

In mathematics, a function[1]
is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x2. The output of a function fcorresponding to an input x is denoted by f(x) (read "f of x"). In this example, if the input is −3, then the output is 9, and we may write f(−3) = 9. The input variable(s) are sometimes referred to as the argument(s) of the function.
Functions of various kinds are "the central objects of investigation"[2] in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function. In science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of adifferential equation.
The input and output of a function can be expressed as an ordered pair, ordered so that the first element is the input (or tuple of inputs, if the function takes more than one input), and the second is the output. In the example above, f(x) = x2, we have the ordered pair (−3, 9). If both input and output are real numbers, this ordered pair can be viewed as the Cartesian coordinates of a point on the graph of the function. But no picture can exactly define every point in an infinite set.
In modern mathematics,[3] a function is defined by its set of inputs, called the domain; a set containing the set of outputs, and possibly additional elements, as members, called its codomain; and the set of all input-output pairs, called its graph. (Sometimes the codomain is called the function's "range", but warning: the word "range" is sometimes used to mean, instead, specifically the set of outputs. An unambiguous word for the latter meaning is the function's "image". To avoid ambiguity, the words "codomain" and "image" are the preferred language for their concepts.) For example, we could define a function using the rule f(x) = x2 by saying that the domain and codomain are the real numbers, and that the graph consists of all pairs of real numbers (xx2). Collections of functions with the same domain and the same codomain are called function spaces, the properties of which are studied in such mathematical disciplines asreal analysiscomplex analysis, and functional analysis.
In analogy with arithmetic, it is possible to define addition, subtraction, multiplication, and division of functions, in those cases where the output is a number. Another important operation defined on functions is function composition, where the output from one function becomes the input to another function.

Introduction and examples


A function that associates to any of the four colored shapes its color.
For an example of a function, let X be the set consisting of four shapes: a red triangle, a yellow rectangle, a green hexagon, and a red square; and let Y be the set consisting of five colors: red, blue, green, pink, and yellow. Linking each shape to its color is a function from Xto Y: each shape is linked to a color (i.e., an element in Y), and each shape is "linked", or "mapped", to exactly one color. There is no shape that lacks a color and no shape that has two or more colors. This function will be referred to as the "color-of-the-shape function".
The input to a function is called the argument and the output is called the value. The set of all permitted inputs to a given function is called the domain of the function, while the set of permissible outputs is called the codomain. Thus, the domain of the "color-of-the-shape function" is the set of the four shapes, and the codomain consists of the five colors. The concept of a function does not require that every possible output is the value of some argument, e.g. the color blue is not the color of any of the four shapes in X.
A second example of a function is the following: the domain is chosen to be the set of natural numbers (1, 2, 3, 4, ...), and the codomain is the set of integers (..., −3, −2, −1, 0, 1, 2, 3, ...). The function associates to any natural number n the number 4−n. For example, to 1 it associates 3 and to 10 it associates −6.
A third example of a function has the set of polygons as domain and the set of natural numbers as codomain. The function associates a polygon with its number of vertices. For example, a triangle is associated with the number 3, a square with the number 4, and so on.
The term range is sometimes used either for the codomain or for the set of all the actual values a function has. To avoid ambiguity this article avoids using the term.

Definition

The above diagram represents a function with domain \{ 1, 2, 3 \}, codomain  \{ A, B, C, D \}  and set of ordered pairs  \{ (1,D), (2,C), (3,C) \} . The image is \{C,D\}.



However, this second diagram does not represent a function. One reason is that 2 is the first element in more than one ordered pair. In particular, (2, B) and (2, C) are both elements of the set of ordered pairs. Another reason, sufficient by itself, is that 3 is not the first element (input) for any ordered pair. A third reason, likewise, is that 4 is not the first element of any ordered pair.
In order to avoid the use of the informally defined concepts of "rules" and "associates", the above intuitive explanation of functions is completed with a formal definition. This definition relies on the notion of the Cartesian product. The Cartesian product of two sets X and Y is the set of all ordered pairs, written (xy), where x is an element of X and y is an element of Y. The x and the y are called the components of the ordered pair. The Cartesian product of X and Y is denoted by X × Y.
A function f from X to Y is a subset of the Cartesian product X × Y subject to the following condition: every element of X is the first component of one and only one ordered pair in the subset.[4] In other words, for every x in X there is exactly one element y such that the ordered pair (x,y) is contained in the subset defining the function f. This formal definition is a precise rendition of the idea that to each x is associated an element y of Y, namely the uniquely specified element y with the property just mentioned.
Considering the "color-of-the-shape" function above, the set X is the domain consisting of the four shapes, while Y is the codomain consisting of five colors. There are twenty possible ordered pairs (four shapes times five colors), one of which is
("yellow rectangle", "red").
The "color-of-the-shape" function described above consists of the set of those ordered pairs,
(shape, color)
where the color is the actual color of the given shape. Thus, the pair ("red triangle", "red") is in the function, but the pair ("yellow rectangle", "red") is not.

Notation[edit]

For more details on this topic, see functional notation.
A function f with domain X and codomain Y is commonly denoted by
f\colon X \rightarrow Y
or
X \stackrel f \rightarrow Y.
In this context, the elements of X are called arguments of f. For each argument x, the corresponding unique y in the codomain is called the function value at x or the image of x under f. It is written as f(x). One says that f associates y with x or maps x to y. This is abbreviated by
y = f(x).
A general function is often denoted by f. Special functions have names, for example, the signum function is denoted by sgn. Given a real number x, its image under the signum function is then written as sgn(x). Here, the argument is denoted by the symbol x, but different symbols may be used in other contexts. For example, in physics, the velocity of some body, depending on the time, is denoted v(t). The parentheses around the argument may be omitted when there is little chance of confusion, thus: sin x; this is known as prefix notation.
In order to denote a specific function, the notation \mapsto (an arrow with a bar at its tail) is used. For example, the above function reads
\begin{align}
 f\colon \mathbb{N} &\to \mathbb{Z} \\
 x &\mapsto 4-x.
\end{align}
The first part can be read as:
  • "f is a function from \mathbb{N} (the set of natural numbers) to \mathbb{Z} (the set of integers)" or
  • "f is a \mathbb{Z}-valued function of an \mathbb{N}-valued variable".
The second part is read:
  • "x maps to 4−x."
In other words, this function has the natural numbers as domain, the integers as codomain. Strictly speaking, a function is properly defined only when the domain and codomain are specified. For example, the formula f(x) = 4 − x alone (without specifying the codomain and domain) is not a properly defined function. Moreover, the function
\begin{align}
 g\colon \mathbb{Z} &\to \mathbb{Z} \\
 x &\mapsto 4-x.
\end{align}
(with different domain) is not considered the same function, even though the formulas defining f and g agree, and similarly with a different codomain. Despite that, many authors drop the specification of the domain and codomain, especially if these are clear from the context. So in this example many just write f(x) = 4 − x. Sometimes, the maximal possible domain is also understood implicitly: a formula such as f(x)=\sqrt{x^2-5x+6} may mean that the domain of f is the set of real numbers x where the square root is defined (in this case x ≤ 2 or x ≥ 3).[5]
To define a function, sometimes a dot notation is used in order to emphasize the functional nature of an expression without assigning a special symbol to the variable. For instance, \scriptstyle a(\cdot)^2 stands for the function \textstyle x\mapsto ax^2\scriptstyle \int_a^{\, \cdot} f(u)du stands for the integral function \scriptstyle x\mapsto \int_a^x f(u)du, and so on.


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