Mathematical Function

Website Links For Function

Information About Mathematical Function

SHOPPER'S DELIGHT

Shop For:   Department All Departments Apparel Automotive Baby Beauty Books Classical Digital Music DVD's Electronics Gourmet Food Personal Care Home & Garden Jewelry Kitchen Music Musical Music Tracks Office Products Outdoor Living PC Hardware Pet Supplies Software Sporting Goods Tools Toys Video Video Games Wireless Wireless Price Minimum: Price Maximum:

In mathematics, a function relates each of its inputs to exactly one output. A standard notation for the output of the function ''f'' with the input ''x'' is ''f''(''x''). The set of all inputs which a function accepts is called the Domain of the function. The set of all outputs is called the Range . For example, the expression ''f''(''x'') = ''x''2 describes a function, ''f'', that relates each input, ''x'', with one output, ''x''2. Thus, an input of 3 is related to an output of 9. Once a function, ''f'', has been defined we can write, for example, ''f''(4) = 16. It is a usual practice in mathematics to introduce functions with temporary names like ''f''; in the next paragraph we might define ''f''(''x'') = 2''x''+1, and then ''f''(4) = 9. When a name for the function is not needed, often the form ''y=x''2 is used. If we use a function often, we may give it a permanent name as, for example :$\backslash mathrm\{Square\}(x)\backslash ,\; =\backslash ,\; x^2$. The essential property of a function is that for each input there must be one unique output. Thus, for example, :$\backslash mathrm\{Root\}(x)\; =\; \backslash pm\; \backslash sqrt\; x$ does not define a function, because it may have two outputs. The square roots of 9 are 3 and −3, for example. To make the square root a function, we must specify which square root to choose. The definition :$\backslash mathrm\{Posroot\}(x)\; =\; \backslash sqrt\; x$, for any non-negative input chooses the non-negative square root as an output. A function need not involve numbers. An example of a function that does not use numbers is the function that assigns to each nation its capital. In this case $Capital(France)=Paris$. A more precise, but still informal definition follows. Let ''A'' and ''B'' be Set s. A function from ''A'' to ''B'' is determined by any association of a unique element of ''B'' with each element of ''A''. The set ''A'' is called the domain of the function; the set ''B'' is called the Codomain . In some contexts, such as Lambda Calculus , the function notion may be taken as primitive rather than defined in terms of set theory. In most mathematical fields, the terms '' Map '', ''mapping'', and '' Transformation '' are usually synonymous with ''function''. However, in some contexts they may be defined with a more specialized meaning. For example, in Topology a map is sometimes defined to be a Continuous Function . MATHEMATICAL DEFINITION OF A FUNCTION A precise definition is required for the purposes of mathematics. A function is a Binary Relation , ''f'', with the property that for an element ''x'' there is no more than one element ''y'' such that ''x'' is related to ''y''. This uniquely determined element ''y'' is denoted by ''f''(''x''). Because two definitions of Binary Relation are in use, there are actually two definitions of function, in effect. First definition The simplest definition of Binary Relation is "A binary relation is a set of Ordered Pair s". Under this definition, the binary relation denoted by "less than" contains the ordered pair (2, 5) because 2 is less than 5. A function is then a set of ordered pairs with the property that if (''a'',''b'') and (''a'',''c'') are in the set, then ''b'' must equal ''c''. Thus the squaring function contains the pair (3, 9). The square root relation is not a function because it contains both the pair (9, 3) and the pair (9, −3), and 3 is not equal to −3. The domain of a function is the set of elements ''x'' occurring as first coordinate in a pair of the relation. If ''x'' is not in the domain of ''f'', then ''f''(''x'') is not defined. The range of a function is the set of elements ''y'' occurring as second coordinate in a pair of the relation. Second definition Some authors require that the definition of a binary relation specify not only the ordered pairs but also the domain and codomain. These authors define a binary relation as an ordered triple $(X,\; Y,\; G)$, where ''X'' and ''Y'' are sets (called the domain and '''codomain''' of the relation) and ''G'' is a subset of the Cartesian Product of ''X'' and ''Y'' (''G'' is called the '''graph''' of the relation). A '''function''' is then a binary relation with the additional property that each element of ''X'' occurs exactly once as the first coordinate of an element of ''G''. Under this second definition, a function has a uniquely determined codomain; this is not the case under the first definition. This form of definition is essential in many contexts. For example, it is impossible to determine if a function is " Onto " without a specified codomain. HISTORY OF THE CONCEPT As a mathematical term, "function" was coined by Gottfried Leibniz in 1694 , to describe a quantity related to a Curve , such as a curve's Slope at a specific Point of a curve. The functions Leibniz considered are today called Differentiable Functions , and they are the type of function most frequently encountered by nonmathematicians. For this type of function, one can talk about Limit s and Derivative s; both are measurements of the output or the change in the output as it depends on the input or the change in the input. Such functions are the basis of Calculus . The word function was later used by Leonhard Euler during the mid- 18th Century to describe an Expression or formula involving various Argument s, e.g. ''f''(''x'') = sin(''x'') + ''x''3. During the 19th Century , mathematicians started to formalize all the different branches of mathematics. Weierstrass advocated building calculus on Arithmetic rather than on Geometry , which favoured Euler's definition over Leibniz's (see Arithmetization Of Analysis ). At first, the idea of a function was rather limited. Joseph Fourier , for example, claimed that every function had a Fourier Series , something no mathematician would claim today. By broadening the definition of functions, mathematicians were able to study "strange" mathematical objects such as continuous functions that are nowhere differentiable. These functions were first thought to be only theoretical curiosities, and they were collectively called "monsters" as late as the turn of the 20th century. However, powerful techniques from Functional Analysis have shown that these functions are in some sense "more common" than differentiable functions. Such functions have since been applied to the modeling of physical phenomena such as Brownian Motion . Towards the end of the 19th century, mathematicians started to formalize all of mathematics using Set Theory , and they sought to define every mathematical object as a Set . Dirichlet and Lobachevsky independently and almost simultaneously gave the modern "formal" definition of function. In this definition, a function is a special case of a Relation , in particular a function is a relation in which every first element has a unique second element. The notion of function as a rule for Computing , rather than a special kind of relation, has been formalized in Mathematical Logic and Theoretical Computer Science by means of several systems, including the Lambda Calculus , the theory of Recursive Function s and the Turing Machine . FUNCTIONS IN OTHER FIELDS Functions are used in every quantitative Science , to model relationships between all kinds of physical quantities — especially when one quantity is completely determined by another quantity. Thus, for example, one may use a function to describe how the Temperature of water affects its Density . Functions are also used in Computer Science to model Data Structure s and the effects of Algorithm s. However, the word is also used in computing in the very different sense of ''procedure'' or ''sub-routine''; see Function (computer Science) . THE VOCABULARY OF FUNCTIONS A specific input value to a function is called an argument of the function. For each argument ''x'', the corresponding unique ''y'' in the codomain is called the function '''value''' at ''x'', or the '''s and most Programming Language s require parentheses around the argument. In some branches of mathematics, such as Automata Theory , the notation ''x'' ''f'' is used instead of f(x). The notation ''f''$\backslash ;:\backslash ;$''x''$\backslash mapsto\{\}$''y'' (''x'' maps to ''y'') is sometimes used to mean ''f''(''x'') = ''y''. The Graph Of A Function ''f'' is the set of all Ordered Pair s (''x'', ''f''(''x'')), for all ''x'' in the domain ''X''. If ''X'' and ''Y'' are subsets of R, the real numbers, then this definition coincides with the familiar sense of "graph" as a picture or plot of the function, with the ordered pairs being the Cartesian Coordinates of points. The concept of the ''image'' can be extended from the image of a point to the image of a Set . If ''A'' is any subset of the domain, then ''f''(''A'') is the subset of the range consisting of all images of elements of A. We say the ''f''(''A'') is the image of A under f. This extension is consistent as long as no subset of the domain is also an element of the domain. A few authors write ''f'' instead of ''f''(''A''), to emphasize the distinction between the two concepts; a few others write ''f''` ''x'' instead of ''f''(''x''), and ''f''``''A'' instead of ''f''[''A'' . Notice that the range of ''f'' is the image ''f''(''X'') of its domain, and that the range of ''f'' is a subset of its codomain. The '' Preimage '' (or ''inverse image'') of a subset ''B'' of the codomain ''Y'' under a function ''f'' is the subset of the domain ''X'' defined by
	:''f''<sup>&nbsp&minus1</sup>(''b'')&nbsp	{''x'' in ''X'' ''f''(''x'') = ''b''}
	The Set Of All Functions From A Set ''X'' To A Set ''Y'' Is Denoted By ''X'' &rarr ''Y'', By	"''X''" class="copylinks" target="_blank">&rarr ''Y'' , or by ''Y''<sup>''X''</sup> The latter notation is justified by the fact that ''Y''<sup>''X''</sup> = ''Y''<sup>''X''</sup> See the article on Cardinal Number s for more details

In some contexts, a relation that satisfies condition (1), but not necessarily (2), may be called a Multivalued Function ; and a relation that satisfies condition (2), but not necessarily (1), may be called a ''' Partial Function '''.


Other properties

There are many other special classes of functions that are important to particular branches of mathematics, or particular applications.
Here is a partial list:

• Continuous , Differentiable , Integrable


• Linear , Polynomial , Rational


• Convex , Monotonic , Unimodal


• Trigonometric


• Algebraic


• Transcendental


• Fractal Function


• Odd Or Even


• Vector -valued.

RESTRICTIONS AND EXTENSIONS

Informally, a Restriction of a function ''f'' is the result of trimming its domain.