Calculus: Analytic Geometry and Calculus, with Vectors

3.1 Functional notation 115

when someone says that the temperature u at the north pole of our earth

is a function of the time t and, without bothering to introduce a new

letter whose significance must be remembered, uses the symbol u(t) to

denote the temperature at time t. Many

problems in pure and applied mathematics

involve functions about which we have some

information and seek more. Moreover, we

must allow ourselves freedom to use standard

terminology that everyone else uses to convey a x, x.. b x

ideas and information. We say that a func-

tion f is increasing over an interval a 5 x 5 b

Figure 3.163

if, as Figure 3.163 indicates, f(xi) < f(x2) whenever a 5 x1 < x2 < b.

Similarly, f is decreasing over the interval if

(3.164) f(xi) > f(x2) (a < x1 < x2 < b).

In this displayed statement, the "whenever" is omitted. The line can be
read "f (xl) > f (X2) whenever a < x1 < x2 < b." If, as Figure 3.163
indicates, f is increasing over the interval a 5 x < b and if f(a) = .1 and
f(b) = B, we say thatf(x) increases from 14 to B as x increases from a to b.
While we use this convenient terminology, we need not be gullible people
who are easily persuaded that numbers x and f(x) can actually increase.
To see 6 increase and say hello to 7 as it proceeds toward 8 could be quite
amusing, but we make no pretense that such things actually happen. To
avoid misunderstandings, the author wishes to publicly proclaim that
he is not recommending rejection of the good old terminology; he is
merely insisting that we know what we mean when we say that y or f(x)
increases as x increases from a to b.
Problem 15 at the end of this section deals with a famous number-
theoretic function. From some points of view, a perfect definition of this
function can be phrased as follows. Let it be the function whose domain
is the set of real numbers and which is such that, for each x in the domain,
rr(x) is the number of primes less than or equal to x. This makes the "law"
or "rule" concept sound very good. We can easily make the pretense
that a sufficiently dynamic operator could produce the numbers 7r(x) that
we need to form the set S of pairs (x,,7r(x)) needed for the static concept.
It will be observed that, in Problem 15, the function is defined in fewer
words.
Trigonometric functions and polynomials are simpler examples of func-
tions that are important in advanced as well as in elementary science.
A polynomial (or polynomial in x) is a function P having values defined by

(3.17) P(x) = aoxn + alxn-1 + + an_1x + a,.

or by

(3.171) P(x) = bo + bix + b2x2 + + bnxn,