Calculus: Analytic Geometry and Calculus, with Vectors

114 Functions, limits, derivatives

The contraption in the central part of Figure 3.151 is guaranteed to
make nearly everybody imagine a more or less complicated process by
which f might operate upon a given input x (an element of the domain of
f) to produce the corresponding output y (an element of the range of f).
The last problem of this section provides ideas about functions, operators,
and transformers that are needed in advanced mathematics and are help-
ful in elementary mathematics.

Figure 3.151

If we know that y is always positive and that x and y are always related

by the formula x2 + y2 = 9, we can discover that y = 1/9- x2 when

-3 < x < 3. Thus y is determined as a function of x which is defined

over the interval -3 < x < 3, and the graph is as shown in Figure 3.16.

Similarly, if we know that y is negative and x2 + y2 = 9, we can conclude

that y = - -\/9 _-x2 and we have a function whose graph appears in

Figure 3.16 Figure 3.161 Figure 3.162

Figure 3.161. If we know that x2 + y2 = 9 but do not know whether y

is positive o. negative, we cannot determine y in terms ofx. The best

we can do is say that, for each x in the interval -3 < x < 3, y is one or

the other of 1/9 - x2 and - N/'9-- x2. Figure 3.162 shows the graph

of a function f for which x2 + [f(x)J2 = 1, it being true that f(x) > 0 for

some values of x and f (x) < 0 for other values of x. Observe that the

equation x2 + y2 = 1 does not, by itself, determiney as a function of x,

but that there do exist functions f for which x2 + [f(x)]2 = 1.

One purpose of all this discussion is to emphasize the fact thatour ideas

about functions must be both broad and precise. Wemust remain calm