8.1 FUNCTIONS AND MAPPINGS 257f (x,) implies x, = x, for all x,, x, E dorn f ." For example, the contra-
positive of the definition states "for all x, and x,, if x, # x,, then f (x,) #
f (x,)." A translation of this condition into plain English yields the require-
ment that distinct x values have distinct corresponding y values. As we
saw earlier, a set of ordered pairs is a function if and only if no two distinct
ordered pairs have the same first element. Analogously, a function, viewed
as a set of ordered pairs, is one to one if and only if no two distinct ordered
pairs have the same second element. For if (x,, y) and (x,, y) are both in
f, for some pair of distinct objects x, and x,, then we have y = f(x,) and
y = f(x,) so that f (x,) = f (x,), but x, # x,, contradicting the definition of
one to one. For real-valued functions of a real variable, the property of
one to one is equivalent to the requirement "each horizontal line y = k
intersects the graph of f in at most one point." (Why? Note also, in this
context, that the definition of function requires that each vertical line meet
the graph in at most one point.) These considerations lead to the following
definition.
DEFINITION 3
A mapping f: A -, B is said to be one to one, or injective, if and only if the func-
tion f is a one-to-one function; that is, whenever x,, x, E A with f(x,) = f(x,), then
x, = x,. Such a mapping is also said to be an injection of A into B.The mapping k: R -, R defined by k(x) = x4 is not injective since 16 =
k(2) = k(- 2), but any linear mapping f: R -, R, f (x) = Mx + B, is an injec-
tion provided M # 0. The one-to-one property of a mapping depends en-
tirely on the "ordered pair part" of the mapping, that is, it is not affected
by the choice of codomain. The same is not true of the companion concept
of onto mapping, which is discussed in the next article.NEW FUNCTIONS FROM OLD ONES
From precalculus courses you are probably familiar with various methods
of constructing functions from given functions. Iff and g both map a subset
of R into R, then the functions f + g, f - g, fg, and f/g also map sub-
sets of R into R. The first three have domain equal to dorn f n dorn g,
while dorn (f /g) = dorn f n dorn g n {x E R I g(x) # 0). The defining rule
for fg, for example, is (fg)(x) = f (x)g(x) for a11 x E dorn f n dorn g, with the
defining rules for f + g, f - g, and f/g given similarly. If k is a real number,
kf is defined by (kf )(x) = kf (x), with dorn (kf) = dorn f. The sum, differ-
ence, product, and quotient of functions are often referred to as operations
on functions.
We now wish to consider three other methods of constructing new func-
tions and mappings from old ones, that is, three other operations on
functions. Unlike the operations of the previous paragraph, which are
dependent for their effectiveness on our ability to add, multiply (and so on)