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),
 
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