8.1 FUNCTIONS AND MAPPINGS 253DEFINITION 1
A function is a relation R having the property that if (x, y) E R and (x, z) E
y = z.Following custom, we will most often use lower-case letters f, g,
so on, rather than R, to denote relations that are functions. If RR, thenh, and
is any
relation, then by Definition 4, Article 7.1, for each x E dorn R, there exists
at least one object y E rng R such that (x, y) E R. Iff is a function, then by
Definition 1 for each x E dorn f, there exists at most one object y E rng f
such that (x, y) E f. In other words, every element x in the domain of a
function f has a unique corresponding y in the range off such that (x, y) E f.
Like any other relation, and more generally like any other set, a function
f may be described by the roster method, that is, by listing all the ordered
pairs, or by the rule method. The former, of course, is applicable only when
f has finitely many ordered pairs; indeed, it is practical only when f has
a relatively small number of ordered pairs. Sometimes, the "pattern" ap-
proach to describing a set may be used to define functions having a finite,
but large, number of ordered pairs.
EXAMPLE 1 The relations f, = ((2, 3), (3, 5), (4, 7), (5,9)), f2 = ((a, z),
(b, Y),... , (Y, b), 6, a)), f3 = (( 1, I), (2,2),... , (100, 100)) are functions,
whereas R, = ((1, a), (1, b),... , (1, z)] and R2 = ((1, 1), (1, - 1), (4,2),
(4, - 2), (9, 3), (9, - 3)) are not functions.A relation described by listing all its ordered pairs is a function if and
only if no two distinct ordered pairs in the list have the same first element.
This criterion could be used in general as a somewhat less precise defini-
tion of the function concept. You should determine the domain and range
of each of the relations in Example 1 [Exercise l(a)]. Also, you should
examine each of the relations in Example 1, Article 7.1, to determine which
are functions [Exercise l(b)].
Iff is a function, then each x E dorn f can be viewed as determining a
unique corresponding y E rng f. For this reason we often refer to this y as
the value of the function f at x, or simply "f of x," and write y = f(x),
rather than (x, y) E f or x f y. In addition, when f is a function, the set
rng f is sometimes referred to as the image off, denoted im f.
When the rule method is used to describe a function, the rule is usually
one that specifies a relationship between each x E dorn f and its corre-
sponding y, such as y = f (x) = x3, y = g(x) = sin- ' x, or "y is the biological
father of x." Thus functions are most often described by designating the
domain and specifying such a rule, often called the rule of correspondence.
When a function is defined by a rule of correspondence y = f(x), the latter
is often referred to as functional notation, in contrast to ordered pair nota-
tion. We can express in these terms what is meant by equality of functions;
two functions f and g are equal if and only if (1) dorn f = dorn g and