<s>
In	O
mathematics	O
,	O
the	O
codomain	B-Algorithm
or	O
set	O
of	O
destination	O
of	O
a	O
function	O
is	O
the	O
set	O
into	O
which	O
all	O
of	O
the	O
output	O
of	O
the	O
function	O
is	O
constrained	O
to	O
fall	O
.	O
</s>
<s>
The	O
term	O
range	B-Algorithm
is	O
sometimes	O
ambiguously	O
used	O
to	O
refer	O
to	O
either	O
the	O
codomain	B-Algorithm
or	O
image	O
of	O
a	O
function	O
.	O
</s>
<s>
A	O
codomain	B-Algorithm
is	O
part	O
of	O
a	O
function	O
if	O
is	O
defined	O
as	O
a	O
triple	O
where	O
is	O
called	O
the	O
domain	B-Algorithm
of	O
,	O
its	O
codomain	B-Algorithm
,	O
and	O
its	O
graph	B-Application
.	O
</s>
<s>
The	O
set	O
of	O
all	O
elements	O
of	O
the	O
form	O
,	O
where	O
ranges	O
over	O
the	O
elements	O
of	O
the	O
domain	B-Algorithm
,	O
is	O
called	O
the	O
image	O
of	O
.	O
</s>
<s>
The	O
image	O
of	O
a	O
function	O
is	O
a	O
subset	O
of	O
its	O
codomain	B-Algorithm
so	O
it	O
might	O
not	O
coincide	O
with	O
it	O
.	O
</s>
<s>
Namely	O
,	O
a	O
function	O
that	O
is	O
not	O
surjective	B-Algorithm
has	O
elements	O
in	O
its	O
codomain	B-Algorithm
for	O
which	O
the	O
equation	O
does	O
not	O
have	O
a	O
solution	O
.	O
</s>
<s>
A	O
codomain	B-Algorithm
is	O
not	O
part	O
of	O
a	O
function	O
if	O
is	O
defined	O
as	O
just	O
a	O
graph	B-Application
.	O
</s>
<s>
For	O
example	O
in	O
set	O
theory	O
it	O
is	O
desirable	O
to	O
permit	O
the	O
domain	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
to	O
be	O
a	O
proper	O
class	O
,	O
in	O
which	O
case	O
there	O
is	O
formally	O
no	O
such	O
thing	O
as	O
a	O
triple	O
.	O
</s>
<s>
With	O
such	O
a	O
definition	O
functions	O
do	O
not	O
have	O
a	O
codomain	B-Algorithm
,	O
although	O
some	O
authors	O
still	O
use	O
it	O
informally	O
after	O
introducing	O
a	O
function	O
in	O
the	O
form	O
.	O
</s>
<s>
the	O
codomain	B-Algorithm
of	O
is	O
,	O
but	O
does	O
not	O
map	O
to	O
any	O
negative	O
number	O
.	O
</s>
<s>
While	O
and	O
map	O
a	O
given	O
to	O
the	O
same	O
number	O
,	O
they	O
are	O
not	O
,	O
in	O
this	O
view	O
,	O
the	O
same	O
function	O
because	O
they	O
have	O
different	O
codomains	B-Algorithm
.	O
</s>
<s>
The	O
domain	B-Algorithm
of	O
cannot	O
be	O
but	O
can	O
be	O
defined	O
to	O
be	O
:	O
</s>
<s>
For	O
this	O
reason	O
,	O
it	O
is	O
possible	O
that	O
,	O
when	O
composed	O
with	O
,	O
might	O
receive	O
an	O
argument	O
for	O
which	O
no	O
output	O
is	O
defined	O
–	O
negative	O
numbers	O
are	O
not	O
elements	O
of	O
the	O
domain	B-Algorithm
of	O
,	O
which	O
is	O
the	O
square	O
root	O
function	O
.	O
</s>
<s>
Function	B-Application
composition	I-Application
therefore	O
is	O
a	O
useful	O
notion	O
only	O
when	O
the	O
codomain	B-Algorithm
of	O
the	O
function	O
on	O
the	O
right	O
side	O
of	O
a	O
composition	O
(	O
not	O
its	O
image	O
,	O
which	O
is	O
a	O
consequence	O
of	O
the	O
function	O
and	O
could	O
be	O
unknown	O
at	O
the	O
level	O
of	O
the	O
composition	O
)	O
is	O
a	O
subset	O
of	O
the	O
domain	B-Algorithm
of	O
the	O
function	O
on	O
the	O
left	O
side	O
.	O
</s>
<s>
The	O
codomain	B-Algorithm
affects	O
whether	O
a	O
function	O
is	O
a	O
surjection	B-Algorithm
,	O
in	O
that	O
the	O
function	O
is	O
surjective	B-Algorithm
if	O
and	O
only	O
if	O
its	O
codomain	B-Algorithm
equals	O
its	O
image	O
.	O
</s>
<s>
In	O
the	O
example	O
,	O
is	O
a	O
surjection	B-Algorithm
while	O
is	O
not	O
.	O
</s>
<s>
The	O
codomain	B-Algorithm
does	O
not	O
affect	O
whether	O
a	O
function	O
is	O
an	O
injection	O
.	O
</s>
<s>
A	O
second	O
example	O
of	O
the	O
difference	O
between	O
codomain	B-Algorithm
and	O
image	O
is	O
demonstrated	O
by	O
the	O
linear	B-Architecture
transformations	I-Architecture
between	O
two	O
vector	O
spaces	O
–	O
in	O
particular	O
,	O
all	O
the	O
linear	B-Architecture
transformations	I-Architecture
from	O
to	O
itself	O
,	O
which	O
can	O
be	O
represented	O
by	O
the	O
matrices	B-Architecture
with	O
real	O
coefficients	O
.	O
</s>
<s>
Each	O
matrix	O
represents	O
a	O
map	O
with	O
the	O
domain	B-Algorithm
and	O
codomain	B-Algorithm
.	O
</s>
<s>
Some	O
transformations	O
may	O
have	O
image	O
equal	O
to	O
the	O
whole	O
codomain	B-Algorithm
(	O
in	O
this	O
case	O
the	O
matrices	B-Architecture
with	O
rank	O
)	O
but	O
many	O
do	O
not	O
,	O
instead	O
mapping	O
into	O
some	O
smaller	O
subspace	O
(	O
the	O
matrices	B-Architecture
with	O
rank	O
or	O
)	O
.	O
</s>
<s>
which	O
represents	O
a	O
linear	B-Architecture
transformation	I-Architecture
that	O
maps	O
the	O
point	O
to	O
.	O
</s>
<s>
The	O
point	O
is	O
not	O
in	O
the	O
image	O
of	O
,	O
but	O
is	O
still	O
in	O
the	O
codomain	B-Algorithm
since	O
linear	B-Architecture
transformations	I-Architecture
from	O
to	O
are	O
of	O
explicit	O
relevance	O
.	O
</s>
<s>
Just	O
like	O
all	O
matrices	B-Architecture
,	O
represents	O
a	O
member	O
of	O
that	O
set	O
.	O
</s>
<s>
Examining	O
the	O
differences	O
between	O
the	O
image	O
and	O
codomain	B-Algorithm
can	O
often	O
be	O
useful	O
for	O
discovering	O
properties	O
of	O
the	O
function	O
in	O
question	O
.	O
</s>
<s>
For	O
example	O
,	O
it	O
can	O
be	O
concluded	O
that	O
does	O
not	O
have	O
full	O
rank	O
since	O
its	O
image	O
is	O
smaller	O
than	O
the	O
whole	O
codomain	B-Algorithm
.	O
</s>
