FUNCTIONS

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 definition 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.
The distinction between f and f ?x? should be kept in mind. f denotes the function as defined in the first paragraph. y and f ?x? are different 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 ?x? 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 corresponding 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 y2 ¼ x, there are two choices of y for each positive value of x. Asanother example, the pairs ?a; b?, ?a; c?, ?a; d?, and ?a; e? 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 definition or modern presentations. In order that there be no ambiguity, thecalculus and its applications require a single image associated with each domain value. A multiplevalued rule of correspondence gives rise to a collection of functions (i.e., single-valued). Thus, the rule y2 ¼ x is replaced by the pair of rules y ¼ x1=2 and y ¼ x1=2 and the functions they generate through the establishment of domains. (See the following section on graphs for pictorial illustrations.)
EXAMPLES. 1. If to each number in 1 @ x @ 1 we associate a number y given by x2, then the interval
is the domain. The rule y ¼ x2 generates the range1. The totality
is a function f .The functional image of x is given by y ¼ f ?x? ¼ x2. For example, domain as the set of positive integers. The rule is the definition of un, and the range is generatedby this rule. To illustrate, let un ¼ 1As you read this chapter, reviewing Chapter 2 will be very useful, and in particular comparing the corresponding sections.
With each time t after the year 1800 we can associate a value P for the population of the United States. The correspondence between P and t defines a function, say F, and we can write
4. For the present, both the domain and the range of a function have been restricted to sets of real numbers. Eventually this limitation will be removed. To get the flavor for greater generality, think of a map of the world on a globe with circles of latitude and longitude as coordinate curves. Assume there is a rule that corresponds this domain to a range that is a region of a plane endowed with a rectangular Cartesian coordinate system. (Thus, a flat map usable for navigation and other purposes is created.) The points of the domain are expressed as pairs of number and those of the range by pairs ?x; y?. These sets and a rule of correspondence constitute a function whose independent and dependent variables are not single real numbers; rather, they are pairs of real numbers.