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In mathematics,
a
function is
a relation between
a set of
inputs
and
a
set
of
permissible outputs with
the property that each input is related to exactly
one output.
An
example
is
the
function
that
relates
each
real
number x to
its
square
x
2
.
The
output
of
a
function
f
corresponding
to
an
input
x
is
denoted
by
f
(
x
)
(read
f
of
x
In
this
example,
if
the
input
is
?
3
,
then
the
output
is
9
,
and
we
may
write
f
(
?
3
)
?
9
. The input variable(s) are sometimes
referred to as the argument(s) of
the
function.
Functions
are
central
objects
of
investigation
most
fields
of
modern
mathematics.
There
are
many
ways
to
describe
or
represent
a
function.
Some
functions
may
be
defined
by
a
formula or algorithm that
tells
how
to
compute
the
output for a given
input. Others are given by a picture, called the
graph of the function.
In
science,
functions
are
sometimes
defined
by
a
table
that
gives
the
outputs
for
selected
inputs.
A
function
can
be
described
through
its
relationship
with
other
functions,
for
example
as
an inverse
function or
as
a
solution
of
a differential
equation.
The input and
output of a function can be expressed as an
ordered pair, ordered so that
the
first
element
is
the
input
(ortuple of
inputs,
if
the
function
takes
more
than
one
input), and the second
is the output.
In the
example above,
f
(
x
)
?
x
2
, we have the
ordered pair
(
?
3
,
9
)
. If both input and output are real
numbers, this ordered pair can
be
viewed as theCartesian coordinates of a point on
the graph of the function. But no
picture
can
exactly
define
every
point
in
an
infinite
set.
In
modern
mathematics,
a
function is defined by its set of
inputs, called the domain, a set containing the
outputs,
called its codomain, and the
set of all paired input and outputs, called the
graph. For
example,
we
could
define
a
function
using
the
rule
f
(
x
)
?
x
2
by
saying
that
the
domain
and codomain are the real numbers, and that the
ordered pairs are all pairs of
real
numbers
(
x
,
x
2
)
. Collections of functions
with
the same domain
and the same
codomain
are
called function
spaces,
the
properties
of
which
are
studied
in
such
mathematical disciplines as
real analysis and complex analysis.
In
analogy with arithmetic, it is possible to define
addition, subtraction, multiplication,
and
division
of
functions,
in
those
cases
where
the
output
is
a
number.
Another
important
operation
defined
on
functions
is function
composition,
where
the
output
from
one function becomes the input to another
function.
A function is called
injective (or one-to-one, or an injection) if
f
(
a
)
?
f
(
b
< br>)
for any
two
different elements a and b of
the
is
called
surjective (or onto)
if
f
(
X
)
?
Y
. That is, it is surjective if
for every element
y
in the
codomain there is
an
x
in
the
domain
such
that
f
(
x
)
?
y
.
Finally
f
is
called bijective if
it
is
both
injective
and surjective. This nomenclature was introduced
by the Bourbaki group.
The
triangle
and
the
red
rectangle)
are
assigned
the
same
value.
Moreover,
it
is
not
surjective, since the
image of the function contains only three, but not
all five colors
in the codomain.
Graph
The graph of a
function is its set of ordered pairs
F
. This is an abstraction of
the idea
of a graph as a picture
showing the function plotted on a pair of
coordinate axes; for
example,
(
3
,
9
)
, the point above
3
on the
horizontal axis
and to
the
right
of
9
on
the vertical axis, lies
on the graph of
y
?
x
2
.
Formulas and
algorithms
Different
formulas
or
algorithms
may
describe
the
same
function.
For
ins
tance
f
(
x
)
?
(
x
?
1
)(
x
?
1
)
is
exactly
the
same
function
as
f
(
x
)
?
x
2
?
1
< br>.
Furthermore, a function need not
be described by a formula, expression, or
algorithm,
nor need it deal with
numbers at all: the domain and codomain of a
function may be
arbitrary
sets .
One
example
of
a
function
that
acts
on
non-
numeric
inputs
takes
English words as inputs and returns the
first letter of the input word as output.
Restrictions and extensions
Informally,
a restriction of
a
function f is
the
result
of
trimming
its
domain.
More
precisely,
if
S
is
any
subset
of
X
,
the
restriction
of
f
to
S
is
the
function
f
s
from
S
to
Y
such
that
f
s
(
s
)
?
< br>f
(
s
)
for
alls in
S
.
If g is
a
restriction
of
f
, then it is said that
f
is an extension of
g
.
The overriding
of
f
:
X
?
Y
by
g
:
W
?
Y
(also
called overriding
union)
is
an
extension
of g denoted
as
(
f
?
g
)
:
(
X
?
W
)
?
Y
.
Its
graph
is
the
set-theoretical
union of the graphs of
g
and
f
x
w
. Thus, it
relates any element of the domain of
g
to its image
under
g
, and any other
element of the domain of
f
to its image under
f
.
Overriding
is
an
associative
operation;
it
has
the empty
function as
an identity
element.
If
f
x
?
w
and
g
x
?
w
are
pointwise
equal
(e.g.,
the
domains
of
f
and
g
g
are disjoint), then the
union of
f
and g
is defined and is equal to their
overriding
union.
This
definition
agrees
with
the
definition
of
union
for binary
relation.
Real-valued
functions
A
real-valued
function
f
is
one
whose
codomain
is
the
set
of real
numbers or
a subset thereof.
If, in addition, the domain is also a subset of
the reals,
f
is a real
valued function of a real variable. The
study of such functions is called real analysis.
Real-valued
functions
enjoy
so-called
pointwise
operations.
That
is,
given
two
functions
f
,
g
:
X
?
Y
where
Y
is a subset of the reals
(and
X
is an arbitrary set),
their
(pointwise)
sum
f
?
g
and
product
f
?
g
are
functions
with
the
same
domain
and codomain.
Function
spaces
The set of all functions from a
set
X
to a set
Y
is denoted by
X
?
Y
,
by
?
X
?
Y
?
,
or by
Y
X
. The latter
notation is motivated by the fact that, when
X
and
Y
are finite
and
of size
X
and
Y
,
then
the
number
of
functions
X
?
Y
is
Y
X
=
Y
X
.
This is an example of the convention
from enumerative combinatorics that provides
notations for
sets based on their cardinalities. If
X
is infinite and there is
more than
one
element
in
Y
then
there
are uncountably
many functions
from
X
to
Y
,
though
only countably many of them can be
expressed with a formula or algorithm.
Alternative definition of a function
The
above
definition
of
function
from
X
to
Y
is
generally
agreed
on, however
there
are
two
different
ways
a
is
normally
defined
where
the
domain
X
and
codomain
Y
are not
explicitly or implicitly specified. Usually this
is
not
a
problem
as
the
domain
and
codomain
normally
will
be
known.
With
one
definition saying the function defined
by
f
(
x
)
?
x
2
on the reals does not completely
specify
a
function
as
the
codomain
is
not
specified,
and
in
the
other
it
is
a
valid
definition.
In the other
definition a function is defined as a set of
ordered pairs where each first
element
only occurs once. The domain is the set of all the
first elements of a pair and
there is
no explicit codomain separate from the image.
Concepts like surjective have
to be
refined for such functions, more specifically by
saying that a (given) function
is
surjective on a (given) set iff its image (or
range) equals that set.
If
a
function
is
defined
as
a
set
of
ordered
pairs
with
no
specific
codomain,
then
f
:
X
?
Y
indicates
that
f
is a function whose
domain is
X
and whose image
is a subset of
Y
.
Y
may be referred to as the
codomain but then any set including the
image
of
f
is
a
valid
codomain
of
f
.
This
is
also
referred
to
by
saying
that
f
maps
X
into
Y
X
and
Y
may subset the ordered
pairs, e.g. the
function f on
the
real
numbers
such
that
y
?
x
2
when
used
as
in
f
:
?
0
< br>,
4
?
?
?
0
,
4
?
means
the
function
defined
only
on
the
interval
?
0
,
2
?
. With
the
definition of a function
as an ordered triple this would always be
considered a partial
function.
An alternative definition of the
composite function
g
?
f
?
x
?
?
defines it for the set
of
all
x
in the
domain of
f
such that
f
?
x
?
< br> is in the domain of
g
.
Thus the real square
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