Why Read This Book? - Index of

120 Chapter 4 Functions
x
A B
Figure 4.1 Schematic of a function: f : A → B.
x
f
Rng f
f (x)
A B
Figure 4.2 The image of x is not unique.
we say mathematically that the image of x ∈ A must be unique? (See Figure 4.2
for a sketch of what must not happen.) One way to say that the rule for f cannot
produce multiple y-values for a single x is to say
(F2) If y1,y2 ∈ B are such that f(x) = y1 and f(x) = y2, then y1 = y2.
At first glance, F2 might seem like a statement that would be true for any
mapping. After all, if f(x) = y1 and f(x) = y2, then the transitive property of
equality ought to allow us to say y1 = y2. Unfortunately, this way of stating F2
does not reveal potential problems in the way f is defined. Here are two examples
that illustrate how a mapping can fail to be well defined.
Example 4.1.2 It is possible that the rule defining f : A → B can be ambiguous.
For example, let f : R + → R be defined in the following way. For x>0, define
f(x) to be a solution y to the equation y 2 = x. You cannot doubt that this set
of instructions generates a real number f(x) from any x in the domain, so that
y 1
y 2