انت هنا الان : شبكة جامعة بابل > موقع الكلية > نظام التعليم الالكتروني > مشاهدة المحاضرة

FUNCTIONS

A function is composed of a domain set, a range set, and a rule of correspondence that assigns

exactly one element of the range to each element of the domain.

This de?nition of a function places no restrictions on the nature of the elements of the two sets.

However, in our early exploration of the calculus, these elements will be real numbers.                         The rule of

correspondence can take various forms, but in advanced calculus it most often is an equation or a set of

equations.

If the elements of the domain and range are represented by x and y, respectively, and f                   symbolizes

the function, then the rule of correspondence takes the form y ¼ f ًxق.

The distinction between f     and f ًxق should be kept in mind.         f denotes the function as de?ned in the

?rst paragraph.         y  and   f ًxق   are di?erent symbols for the range (or image) values corresponding to

domain values x. However a ‘‘common practice’’ that provides an expediency in presentation is to read

f ًxق as, ‘‘the image of x with respect to the function f ’’ and then use it when referring to the function.

(For example, it is simpler to write sin x than ‘‘the sine function, the image value of which is sin x.’’)

This deviation from precise notation will appear in the text because of its value in exhibiting the ideas.

The domain variable x is called the independent variable.                  The variable y representing the corre-

sponding set of values in the range, is the dependent variable.

Note: There is nothing exclusive about the use of                x,  y, and    f  to represent domain, range, and

function.     Many other letters will be employed.

There are many ways to relate the elements of two sets. [Not all of them correspond a unique range

value to a given domain value.] For example, given the equation y²¼ x, there are two choices of y for

each positive value of x. As another example, the pairs ًa; bق, ًa; cق, ًa; dق, and ًa; eق can be formed and

again the correspondence to a domain value is not unique.                  Because of such possibilities, some texts,

especially older ones, distinguish between multiple-valued and single-valued functions. This viewpoint

is not consistent with our de?nition or modern presentations. In order that there be no ambiguity, the

calculus and its applications require a single image associated with each domain value.                                A multiple-

valued rule of correspondence gives rise to a collection of functions (i.e., single-valued). Thus, the rule

y²¼ x is replaced by the pair of rules y ¼ x¹=2_andy ¼ x¹=2 and the functions they generate through the