Functions
Advanced Level Pure Mathematics
(ii)
f
b
a
b
a
∈
2200
)
,
(
),
,
(
2
2
1
1
, if
2
1
a
a
=
, then
2
1
b
b
=
.
For any
A
a
∈
,
)
(
a
f
is unique.

)
(
a
f
value of
f
at
a
.
f
1
Pr
is the first projection of the ordered pair of
.
f

f
2
Pr
Image of
f

A
Domain of
f

B
Range of
f
Example
Let
{
}
5
,
4
,
3
,
2
,
1
=
A
,
{
}
e
d
c
b
a
B
,
,
,
,
=
The following are functions (mappings) from
A
to
B
.
{
}
)
,
4
(
),
,
3
(
),
,
2
(
),
,
1
(
1
c
b
c
a
f
=
{
}
)
,
4
(
),
,
3
(
),
,
2
(
),
,
1
(
2
a
b
c
d
f
=
The following are not functions (mappings) from
A
to
B
.
{
}
)
,
3
(
),
,
2
(
),
,
1
(
1
c
b
a
g
=
{
}
)
,
4
(
),
,
3
(
),
,
2
(
),
,
1
(
),
,
1
(
2
c
a
a
b
a
g
=
{
}
)
,
4
(
),
,
3
(
),
,
2
(
),
,
1
(
3
h
f
e
d
g
=
Remark
A function of real variable is a function whose domain is the set of all real numbers or a subset
of
R
. A realvalued function is a function whose range is the set of all real numbers.
Example
Let
R
x
∈
, find the domain as long as possible of each of the following functions.
(a)
5
2
)
(

=
x
x
f
(b)
x
x
f
1
)
(
=
(c)
x
x
f
=
)
(
Prepared by
K. F. Ngai
Page
2