Section 2.1 What Is a Function?
DEFINITION: A
function
f
is a rule that assigns to each element
x
in a set
A
exactly one
element, called
f
(
x
)
,
in a set
B
. The set
A
is called the
domain
of
f.
The
range
of
f
is the
set of all possible values of
f
(
x
) as
x
varies throughout the domain.
EXAMPLES:
1. Let
f
(
x
) =
x
+
√
x.
Then
f
(0) = 0 +
√
0 = 0
f
(4) = 4 +
√
4 = 6
2. Let
f
(
x
) = 3
x
2
+
x

5
.
Then
f
(

2) = 3
·
(

2)
2
+ (

2)

5 = 5
f
(0) = 3
·
0
2
+ 0

5 =

5
f
(4) = 3
·
4
2
+ 4

5 = 47
f
(
1
2
)
= 3
·
(
1
2
)
2
+
1
2

5 =

15
4
3. Let
f
(
x
) =
√
x
+ 1
x
. Then
f
(3) =
√
3 + 1
3
=
√
4
3
=
2
3
f
(5) =
√
5 + 1
5
=
√
6
5
and
f
(
a

1) =
√
a

1 + 1
a

1
=
√
a
a

1
4. Let
f
(
x
) =
{
3
x
2
+
x

5 if
x
≤
0
x
+
√
x
if
x >
0
and
g
(
x
) =
{
3
x
2
+
x

5 if
x <
0
x
+
√
x
if
x
≥
0
Then
f
(

2) = 3
·
(

2)
2
+ (

2)

5 = 5
f
(0) = 3
·
0
2
+ 0

5 =

5
f
(4) = 4 +
√
4 = 6
and
g
(

2) = 3
·
(

2)
2
+ (

2)

5 = 5
g
(0) = 0 +
√
0 = 0
g
(4) = 4 +
√
4 = 6
1
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 Spring '12
 KIRYLTSISHCHANKA
 Calculus, Algebra

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