2. INTRODUCTION TO FUNCTIONS
15
2. Introduction to Functions
2.1. Function.
Definition
2.1.1
.
Let
A
and
B
be sets. A
function
f
:
A
→
B
is a rule which assigns to every element in
A
exactly one element in
B
.
If
f
assigns
a
∈
A
to the element
b
∈
B
, then we write
f
(
a
) =
b,
and we call
b
the
image
or
value
of
f
at
a
.
Discussion
This is the familiar deﬁnition of a function
f
from a set
A
to a set
B
as a rule that
assigns each element of
A
to exactly one element
B
. This is probably quite familiar
to you from your courses in algebra and calculus. In the context of those subjects,
the sets
A
and
B
are usually subsets of real numbers
R
, and the
rule
usually refers
to some concatenation of algebraic or transcendental operations which, when applied
to a number in the set
A
, give a number in the set
B
. For example, we may deﬁne a
function
f
:
R
→
R
by the formula (rule)
f
(
x
) =
√
1 + sin
x
. We can then compute
values of
f
– for example,
f
(0) = 1,
f
(
π/
2) =
√
2,
f
(3
π/
2) = 0,
f
(1) = 1
.
357
(approximately) – using knowledge of the sine function at special values of
x
and/or
a calculator. Sometimes the rule may vary depending on which part of the set
A
the
element
x
belongs. For example, the absolute value function
f
:
R
→
R
is deﬁned by
f
(
x
) =

x

=
±
x,
if
x
≥
0
,

x,
if
x <
0
.
The
rule
that deﬁnes a function, however, need not be given by a formula of such
as those above. For example, the rule that assigns to each resident of the state of
Florida his or her last name deﬁnes a function from the set of Florida residents to
the set of all possible names. There is certainly nothing formulaic about the rule
that deﬁnes this function. At the extreme we could randomly assign everyone in this
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 Fall '08
 PenelopeKirby
 Rational number, codomain

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