Graphics Reference
In-Depth Information
The range and co-domain of a function
The set of possible values of
f
(
x
) is called the
range
of thefunction,
and any set which contains the range may be declared to be the
co
-
domain
of thefunction. Wehavejust given a function with domain
Z
and range N
0
, which we express symbolically by writing
f
: Z
N
0
, with thedefinition
f
(
x
)
x
, (or,
f
:
x
x
).
Thetrms
function
and
mapping
are synonymous, and we sometimes
say that thefunction
f
maps
x
to
f
(
x
). In fact each of the sequences in
chapter 3 is a function with domain N. When both the domain and
co-domain of
f
aresubsts of
R, thefunction
f
is called a
real function
.
Arrow diagram of the function f
Domain of ƒ = {values of
x
}
1
0
1
2
3
4
n
n
+ 1
2
ƒ(
x
) =
x
2
0
1
2
3
4
n
Co-domain of ƒ range of ƒ = {values of ƒ(
x
)}
2
Thefollowing functions each havedomain and co-domain
R
. What
is therangeof each function?
x
x
(i)
f
(
x
)
, (ii)
f
(
x
)
, (iii)
f
(
x
)
sin
x
,
(iv)
f
(
x
)
1/(1
x
), (v)
f
(
x
)
e
[The sine and exponential functions will be defined formally
in chapter 11.]
When theco-domain and rangeof a function arethesam, the
function is said to be
onto
theco-domain and is called a
surjection
.
Check that only one of the functions in question 2 is
onto
R
.
Thedistinctiveproprty of a function or mapping
f
is that, for a
given element
x
of thedomain,
f
(
x
) is uniquely determined as a member
of theco-domain. Thus
(1
x
) does not define a function of
x
,
unless the domain is restricted to
1
, for then the range is simply
0
. But even
f
(
x
)
(1
x
) does not define a function with domain
and co-domain R, for then
f
(
x
) is not defined when
x
1.
3
Each of thefollowing functions has domain
A
R and co-domain
R. What is thelargst possibledomain
A
for thefunction?
(i)
f
(
x
)
x
, (ii)
f
(
x
)
(1
x
), (iii)
f
(
x
)
(
x
1),