Function.

Ci-dessous un extrait traitant le sujet : Function.. Ce document contient 717 mots. Pour le télécharger en entier, envoyez-nous un de vos documents grâce à notre système d’échange gratuit de ressources numériques ou achetez-le pour la modique somme d’un euro symbolique. Cette aide totalement rédigée en format pdf sera utile aux lycéens ou étudiants ayant un devoir à réaliser ou une leçon à approfondir en : Echange

---

Function.
Function, in mathematics, term used to indicate the relationship or correspondence between two or more quantities. The term function was first used in 1637 by the
French mathematician René Descartes to designate a power xn of a variable x. In 1694 the German mathematician Gottfried Wilhelm Leibniz applied the term to various
aspects of a curve, such as its slope. The most widely used meaning until quite recently was defined in 1829 by the German mathematician Peter Dirichlet. Dirichlet
conceived of a function as a variable y, called the dependent variable, having its values fixed or determined in some definite manner by the values assigned to the
independent variable x, or to several independent variables x1, x2, ..., xk.
The values of both the dependent and independent variables were real or complex numbers. The statement y = f(x), read " y is a function of x," indicated the
interdependence between the variables x and y; f(x) was usually given as an explicit formula, such as f(x) = x2 - 3x + 5, or by a rule stated in words, such as f(x) is the
first integer larger than x for all x's that are real numbers (see Number). If a is a number, then f(a) is the value of the function for the value x = a. Thus, in the first
example, f(3) = 32 - 3 · 3 + 5 = 5, f(-4) = (-4)2 - 3(-4) + 5 = 33; in the second example, f(3) = f(3.1) = f(p) = 4.
The emergence of set theory first extended and then altered substantially the concept of a function. The function concept in present-day mathematics may be
illustrated as follows. Let X and Y be two sets with arbitrary elements; let the variable x represent a member of the set X, and let the variable y represent a member of
the set Y. The elements of these two sets may or may not be numbers, and the elements of X are not necessarily of the same type as those of Y. For example, X might
be the set of the 50 states of the United States and Y the set of positive integers. Let P be the set of all possible ordered pairs (x, y) and F a subset of P with the
property that if (x1, y1) and (x2, y2) are two elements of F, then y1?y2 implies that x1?x2--that is, F contains no more than one ordered pair with a given x as its first
member. (If x1?x2, however, it may happen that y1 = y2.) A function is now regarded as the set F of ordered pairs with the stated condition and is written F:X-> Y. The
set X1 of x's that occur as first elements in the ordered pairs of F is called the domain of the function F; the set Y1 of y's that occur as second elements in the ordered
pairs is called the range of the function F. Thus, {(New York, 7), (Ohio, 4), (Utah, 4)} is one function that has X = the set of the 50 U.S. states and Y = the set of all
positive integers; the domain is the three states named, and the range is 4, 7.
The modern concept of a function is related to the Dirichlet concept. Dirichlet regarded y = x2 - 3x + 5 as a function; today, y = x2 - 3x + 5 is thought of as the rule
that determines the correspondent y for a given x of an ordered pair of the function; thus, the preceding rule determines (3, 5), (-4, 33) as two of the infinite number
of elements of the function. Although y = f(x) is still used today, it is better to read it as "y is functionally related to x".
A function is also called a transformation or mapping in many branches of mathematics. If the range Y1 is a proper subset of Y (that is, at least one y is in Y but not in
Y1), then F is a function or transformation or mapping of the domain X1 into Y; if Y1 = Y,F is a function or transformation or mapping of X1 onto Y.

Contributed By:
James Singer
Reviewed By:
J. Lennart Berggren
Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.