
In
mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the image of a
function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
is the set of all output values it may produce.
More generally, evaluating a given function
at each
element
Element may refer to:
Science
* Chemical element
Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements
In chemistry, an element is a pure substance consisting only of atoms that all ...
of a given subset
of its
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Doma ...
produces a set, called the "image of
under (or through)
". Similarly, the inverse image (or preimage) of a given subset
of the
codomain
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of
is the set of all elements of the domain that map to the members of
Image and inverse image may also be defined for general
binary relations
Binary may refer to:
Science and technology
Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that t ...
, not just functions.
Definition
The word "image" is used in three related ways. In these definitions,
is a
function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
from the
set to the set
Image of an element
If
is a member of
then the image of
under
denoted
is the
value
Value or values may refer to:
* Value (ethics) it may be described as treating actions themselves as abstract objects, putting value to them
** Values (Western philosophy) expands the notion of value beyond that of ethics, but limited to Western s ...
of
when applied to
is alternatively known as the output of
for argument
Given
the function
is said to "" or "" if there exists some
in the function's domain such that
Similarly, given a set
is said to "" if there exists
in the function's domain such that
However, "" and "" means that
for point
in
's domain.
Image of a subset
Throughout, let
be a function.
The under
of a subset
of
is the set of all
for
It is denoted by
or by
when there is no risk of confusion. Using
set-builder notation
In set theory
illustrating the intersection (set theory), intersection of two set (mathematics), sets.
Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ...
, this definition can be written as
This induces a function
where
denotes the
power set
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
of a set
that is the set of all
subset
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s of
See below for more.
Image of a function
The ''image'' of a function is the image of its entire
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Doma ...
, also known as the
range
Range may refer to:
Geography
* Range (geographic)A range, in geography, is a chain of hill
A hill is a landform
A landform is a natural or artificial feature of the solid surface of the Earth or other planetary body. Landforms together ...
of the function. This usage should be avoided because the word "range" is also commonly used to mean the
codomain
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of
Generalization to binary relations
If
is an arbitrary
binary relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of elem ...
on
then the set
is called the image, or the range, of
Dually, the set
is called the domain of
Inverse image
Let
be a function from
to
The preimage or inverse image of a set
under
denoted by
is the subset of
defined by
Other notations include
and
The inverse image of a
singleton set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, denoted by