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LECTURE 20: FUNCTIONS (
§
9.19.2)
Success is more a function of consistent common sense than it is of genius. An Wang
1.
What are functions?
Let
A
and
B
be two nonempty sets. A
function
f
from
A
to
B
, written
f
:
A
→
B
, is a relation
(i.e. a subset of
A
×
B
) where each element of
A
occurs as the ﬁrst coordinate of exactly one ordered
pair. We think of that ordered pair (
a,b
) as telling us where
a
is mapped to:
b
=
f
(
a
).
Example 1.
Let
f
:
R
→
R
be given by the rule
f
(
x
) =
x
2
. We can also think of
f
as the set of
ordered pairs
f
=
{
(
x,y
)
∈
R
×
R
:
y
=
x
2
}
. Notice that each element of
A
=
R
occurs exactly once
as a ﬁrst coordinate.
The set
A
is called the
domain
. The set
B
is called the
codomain
. The set of elements inside
B
which do occur as second coordinates is called the
range
. We usually write dom(
f
) for the domain, and
ran(
f
) for the range.
Example 2.
Let
f
:
R
→
R
be given by the rule
f
(
x
) =
x
2
. The domain is
R
, the codomain is
R
, and
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This note was uploaded on 03/08/2011 for the course MATH 334 taught by Professor Smith during the Spring '11 term at Vanderbilt.
 Spring '11
 Smith
 Sets

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