In mathematics, the image of a function
f
:
X
→
Y
{\displaystyle f:X\to Y}
is the set of all
f
(
x
)
{\displaystyle f(x)}
such that
x
{\displaystyle x}
belongs to the domain of
f
{\displaystyle f}
.
The image by
f
{\displaystyle f}
of an element
x
{\displaystyle x}
of the domain of
f
{\displaystyle f}
is
f
(
x
)
{\displaystyle f(x)}
, that is, the output corresponding to the input
x
{\displaystyle x}
.
The image by
f
{\displaystyle f}
of a subset
S
{\displaystyle S}
of the domain of
f
{\displaystyle f}
is the set of all
f
(
x
)
{\displaystyle f(x)}
such that
x
{\displaystyle x}
is in
S
{\displaystyle S}
, that is, the set of the images of the elements of
S
{\displaystyle S}
. Equivalently, it is the image of the restriction of
f
{\displaystyle f}
to
S
{\displaystyle S}
.
Preimages or inverse images are defined similarly, by exchanging the roles of the domain and the codomain:
The preimage of an element
y
{\displaystyle y}
of the codomain of
f
{\displaystyle f}
is the set of all elements
x
{\displaystyle x}
of the domain of
f
{\displaystyle f}
such that
f
(
x
)
=
y
{\displaystyle f(x)=y}
; it is empty if
y
{\displaystyle y}
does not belong to the image of
f
{\displaystyle f}
. The preimage of a subset
T
{\displaystyle T}
of the codomain of
f
{\displaystyle f}
is the set of all elements
x
{\displaystyle x}
of the domain of
f
{\displaystyle f}
such that
f
(
x
)
∈
T
{\displaystyle f(x)\in T}
. The preimage of the codomain of
f
{\displaystyle f}
is, by definition of a function, the domain of
f
{\displaystyle f}
.
Images and inverse images may also be defined similarly for general binary relations. in this generalization, images and preimages play symmetric roles: the images and the preimages of a relation are respectively the preimages and the images of the opposite relation.
Contents
Definition
The word "image" is used in three related ways. In these definitions,
f
:
X
→
Y
{\displaystyle f:X\to Y}
is a function from the set
X
{\displaystyle X}
to the set
Y
{\displaystyle Y}
.
Image of an element
If
x
{\displaystyle x}
is a member of
X
{\displaystyle X}
, then the image of
x
{\displaystyle x}
under
f
{\displaystyle f}
, denoted
f
(
x
)
{\displaystyle f(x)}
, is the value of
f
{\displaystyle f}
when applied to
x
{\displaystyle x}
.
f
(
x
)
{\displaystyle f(x)}
is alternatively known as the output of
f
{\displaystyle f}
Image of a subset
Throughout, let
f
:
X
→
Y
{\displaystyle f:X\to Y}
be a function.
The image under
f
{\displaystyle f}
of a subset
A
{\displaystyle A}
of
X
{\displaystyle X}
is the set of all
f
(
a
)
{\displaystyle f(a)}
for
a
∈
A
{\displaystyle a\in A}
. It is denoted by
f
[
A
]
{\displaystyle f[A]}
Image of a function
The image of a function is the image of its entire domain, also known as the range of the function. This last usage should be avoided because the word "range" is also commonly used to mean the codomain of
f
{\displaystyle f}
.
Generalization to binary relations
If
R
{\displaystyle R}
is an arbitrary binary relation on
X
×
Y
{\displaystyle X\times Y}
, then the set
{
y
∈
Y
:
x
R
y
for some
x
∈
X
}
{\displaystyle \{y\in Y:xRy{\text{ for some }}x\in X\}}
is called the image, or the range, of
R
{\displaystyle R}
. Dually, the set
{
x
∈
X
:
x
Inverse image
Let
f
{\displaystyle f}
be a function from
X
{\displaystyle X}
to
Y
.
{\displaystyle Y.}
The preimage or inverse image of a set
B
⊆
Y
{\displaystyle B\subseteq Y}
under
f
,
{\displaystyle f,}
denoted by
f
−
1
[
B
]
,
{\displaystyle f^{-1}[B],}
is the subset of
X
{\displaystyle X}
defined by
f
Notation for image and inverse image
The traditional notations used in the previous section do not distinguish the original function
f
:
X
→
Y
{\displaystyle f:X\to Y}
from the image-of-sets function
f
:
P
(
X
)
→
P
(
Y
)
{\displaystyle f:{\mathcal {P}}(X)\to {\mathcal {P}}(Y)}
; likewise they do not distinguish the inverse function (assuming one exists) from the inverse image function (which again relates the powersets). Given the right context, this keeps the notation light and usually does not cause confusion. But if needed, an alternative is to give explicit names for the image and preimage as functions between power sets:
Arrow notation
f
→
:
P
(
X
)
→
P
(
Y
)
{\displaystyle f^{\rightarrow }:{\mathcal {P}}(X)\to {\mathcal {P}}(Y)}
with
f
→
(
A
)
=
{
f
(
a
)
|
a
∈
A
}
{\displaystyle f^{\rightarrow }(A)=\{f(a)\;|\;a\in A\}}
f
←
:
P
(
Star notation
f
⋆
:
P
(
X
)
→
P
(
Y
)
{\displaystyle f_{\star }:{\mathcal {P}}(X)\to {\mathcal {P}}(Y)}
instead of
f
→
{\displaystyle f^{\rightarrow }}
f
⋆
:
P
(
Y
)
→
P
(
X
)
{\displaystyle f^{\star }:{\mathcal {P}}(Y)\to {\mathcal {P}}(X)}
instead of
f
←
{\displaystyle f^{\leftarrow }}
Other terminology
An alternative notation for
f
[
A
]
{\displaystyle f[A]}
used in mathematical logic and set theory is
f
″
A
.
{\displaystyle f\,''A.}
Some texts refer to the image of
f
{\displaystyle f}
as the range of
f
,
{\displaystyle f,}
but this usage should be avoided because the word "range" is also commonly used to mean the codomain of
f
.
{\displaystyle f.}
Examples
f
:
{
1
,
2
,
3
}
→
{
a
,
b
,
c
,
d
}
{\displaystyle f:\{1,2,3\}\to \{a,b,c,d\}}
defined by
{
1
↦
a
,
2
↦
a
,
3
↦
c
.
{\displaystyle \left\{{\begin{matrix}1\mapsto a,\\2\mapsto a,\\3\mapsto c.\end{matrix}}\right.}
Properties
General
For every function
f
:
X
→
Y
{\displaystyle f:X\to Y}
and all subsets
A
⊆
X
{\displaystyle A\subseteq X}
and
B
⊆
Y
,
{\displaystyle B\subseteq Y,}
the following properties hold:
Also:
f
(
A
)
∩
B
=
∅
if and only if
A
∩
f
−
1
(
B
Multiple functions
For functions
f
:
X
→
Y
{\displaystyle f:X\to Y}
and
g
:
Y
→
Z
{\displaystyle g:Y\to Z}
with subsets
A
⊆
X
{\displaystyle A\subseteq X}
and
C
⊆
Z
,
{\displaystyle C\subseteq Z,}
the following properties hold:
(
g
∘
f
)
(
A
)
=
Multiple subsets of domain or codomain
For function
f
:
X
→
Y
{\displaystyle f:X\to Y}
and subsets
A
,
B
⊆
X
{\displaystyle A,B\subseteq X}
and
S
,
T
⊆
Y
,
{\displaystyle S,T\subseteq Y,}
the following properties hold:
The results relating images and preimages to the (Boolean) algebra of intersection and union work for any collection of subsets, not just for pairs of subsets:
f
(
⋃
s
∈
S
A
s