W4 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8(2) f(x) = g(x) for all x in the common domain. You are asked to prove
this fact in Exercise 2(a). The following example provides several functions
defined in the manner just described.EXAMPLE 2 (a) Define g , by dom g, = (x E R 1 x 2 0) and g ,(x) = x2.
(b) Define g2 by dom g2 = R and g,(x) = (x3 + 5)lI3.
(c) Define g3 by dom g, = [4, co) and g3(x) = dm, where &
represents the unique positive square root of x. 0The relations g,, g,, and g3 in Example 2 are all functions, whereas
the relation R defined by the (inappropriate, but common) notation R(x) =
+A, dom R = [0, a), is not a function (Why?).
A common practice, in cases where the domain of a function f is a subset
of the reals, is to describe the function by stating the rule only, with the
understanding that the domain is the set of all real numbers for which
the rule makes sense. With this understanding, specification of the domain
in (b) and (c) of Example 2 is superfluous. On the other hand, part (a) of
Example 2 requires that the domain be given explicitly, since the "squaring
function" is otherwise assumed to have domain R, that is, g, is a different
function from the function we designate simply as f(x) = x2.
Because of the convention discussed in the preceding paragraph, it is
common to think of the rule describing a function as the function. This
interpretation can lead to difficulties, especially since it is possible to have
a function without an explicit rule (see, e.g., Example 4, Article 8.3), but
is consistent with the widespread and very useful practice of considering
functions from a dynamic, rather than a static, point of view. In this con-
text, emphasis is placed on the transformation of an independent variable
(x, representing elements in the domain) into a dependent variable [y, given
by the formula y = f(x), representing elements of the range]. It is impor-
tant that you be able to pass comfortably between the ordered pair ap-
proach and various other ways of interpreting what a function is, so that,
for example, the function described by g(x) = x3 or "the cubing function"
or "the function that sends (or maps) x to x3" or the function described
by the graph in Figure 8.1, can be recognized as ((x, x3) lx E RJ.
In the context of real-valued functions of a real variable certain catego-
ries of function are determined with respect to the form of the defining
rule. An algebraic function is defined by an equation in variables x and
y involving a finite number of algebraic operations, that is, sums, differences,
products, quotients, nth powers, and nth roots. An example is provided
by the function y = [(x3 - 5x2 + 3x + 7)/(x4 - 3x + 2)I3l5. A function
such as f (x) = sin (x) or g(x) = 2x, which can be shown to be nonalgebraic,
is said to be transcendental. Important subcategories of algebraic functions,
listed in order of decreasing generality, are rational functions, polynomial
functions, linear functions, and the identity function. You should already
be familiar with these categories and able to give examples of functions ofL