Calculus: Analytic Geometry and Calculus, with Vectors

3.1 Functional notation 113

positive because radii of disks are positive numbers), we can say that y is a
function of x and write y = g(x). Then g(2) is the area of a disk of radius
2 and g(2.03) is the area of a disk of radius 2.03. In each case g(x) = 7rx2.
We can think of g as being the operator which converts x into 7rx2 when
x > 0 or as being the set of ordered pairs (x,irx2) for which x > 0.
It is important to know about a particular special way in which a scalar
function of one scalar variable can be determined. Suppose we have a
given set S of ordered pairs (x,y) of numbers such that the set does not
contain two pairs (xl,yl) and (x2,y2) for which x2 = x, and 3'2 vi. To
each number xo that appears as the first number in one of the pairs (x,y),
there is then one and only one number yo such that the pair (xo,yo)
appears in the set. We may let f(xo) denote
this number yo, and we have yo = f(xo). Thus
the given set S of ordered pairs (x,y) becomes
the set of ordered pairs (x, f(x)). When the
pairs of numbers in the set are associated with

zI

1 -o

points and are plotted in the usual way, an

the con- O^123 x

example being shown in Figure 3.15,

dition on the ordered pairs means that no two

Figure 3.15

points fall on the same vertical line. In the example of the figure, we

see that f(x) = 2 when x = 0, that f(x) =1 when 1 < x < 2, that

f (x) = 2 when x = 2, and that f (x) = x - 2 when 2 < x < 3. When

x has a value different from 0 and notin the interval 1 < x _<_ 3, no

meaning has been attached to f(x) and we say that f(x) is undefined.

In this and other cases, the set of values of x for which f(x) is defined is

called the domain of the function, and the set of values attained by f (x)

is called the range of the function. All this is perfectly explicit and pre-

cise, and it should be thoroughly understood by everyone. One who

wishes to regard f as an operator must realize that each set S of the type

described above completely determines the number f (x) that f must

produce when it operates upon a given number x in the domain of f.

Likewise, one who wishes to regard f as a set S must realize that the opera-

tor f determines his set S of pairs (x, f(x)). Everyone must realize that a

set S* of points in a plane endowed with an x, y coordinate systemdeter-

mines both the operator f and the set S, provided no two points ofS*

lie on the same vertical line. As sometimes happens in mathematics and

elsewhere, we have a situation in which different individuals canhold

different personal preferences. For example, a person who wishes to

regard f as an operator can take a dim view of the idea that anappropriate

set S of ordered pairs of numbers "is" a function becauseit determines a

function. He can feel that this is too much like saying that a social

security number "is" a worker because it determines a worker, and he can

object to the idea that social security numbers eat mashed potatoes.