In mathematics, the image of a function is the set of all output values it may produce.
More generally, evaluating a given function $f$ at each

$f\; :\; \backslash R\; \backslash to\; \backslash R$ defined by $x\; \backslash mapsto\; x^2,$

showing that equality generally need

not hold for some laws: , - , , - , , - ,

(equal if $B\; \backslash subseteq\; f(X);$ for instance, if $f$ is surjective)See See , $f^\{-1\}(f(A))\; \backslash supseteq\; A$

(equal if $f$ is injective) , - , $f(f^\{-1\}(B))\; =\; B\; \backslash cap\; f(X)$ , $\backslash left(f\; \backslash vert\_A\backslash right)^\{-1\}(B)\; =\; A\; \backslash cap\; f^\{-1\}(B)$ , - , $f\backslash left(f^\{-1\}(f(A))\backslash right)\; =\; f(A)$ , $f^\{-1\}\backslash left(f\backslash left(f^\{-1\}(B)\backslash right)\backslash right)\; =\; f^\{-1\}(B)$ , - , $f(A)\; =\; \backslash varnothing\; \backslash ,\backslash text\{\; if\; and\; only\; if\; \}\backslash ,\; A\; =\; \backslash varnothing$ , $f^\{-1\}(B)\; =\; \backslash varnothing\; \backslash ,\backslash text\{\; if\; and\; only\; if\; \}\backslash ,\; B\; \backslash subseteq\; Y\; \backslash setminus\; f(X)$ , - , $f(A)\; \backslash supseteq\; B\; \backslash ,\backslash text\{\; if\; and\; only\; if\; \}\; \backslash text\{\; there\; exists\; \}\; C\; \backslash subseteq\; A\; \backslash text\{\; such\; that\; \}\; f(C)\; =\; B$ , $f^\{-1\}(B)\; \backslash supseteq\; A\; \backslash ,\backslash text\{\; if\; and\; only\; if\; \}\backslash ,\; f(A)\; \backslash subseteq\; B$ , - , $f(A)\; \backslash supseteq\; f(X\; \backslash setminus\; A)\; \backslash ,\backslash text\{\; if\; and\; only\; if\; \}\backslash ,\; f(A)\; =\; f(X)$ , $f^\{-1\}(B)\; \backslash supseteq\; f^\{-1\}(Y\; \backslash setminus\; B)\; \backslash ,\backslash text\{\; if\; and\; only\; if\; \}\backslash ,\; f^\{-1\}(B)\; =\; X$ , - , $f(X\; \backslash setminus\; A)\; \backslash supseteq\; f(X)\; \backslash setminus\; f(A)$ , $f^\{-1\}(Y\; \backslash setminus\; B)\; =\; X\; \backslash setminus\; f^\{-1\}(B)$ , - , $f\backslash left(A\; \backslash cup\; f^\{-1\}(B)\backslash right)\; \backslash subseteq\; f(A)\; \backslash cup\; B$See p.388 of Lee, John M. (2010). Introduction to Topological Manifolds, 2nd Ed. , $f^\{-1\}(f(A)\; \backslash cup\; B)\; \backslash supseteq\; A\; \backslash cup\; f^\{-1\}(B)$ , - , $f\backslash left(A\; \backslash cap\; f^\{-1\}(B)\backslash right)\; =\; f(A)\; \backslash cap\; B$ , $f^\{-1\}(f(A)\; \backslash cap\; B)\; \backslash supseteq\; A\; \backslash cap\; f^\{-1\}(B)$ Also: * $f(A)\; \backslash cap\; B\; =\; \backslash varnothing\; \backslash ,\backslash text\{\; if\; and\; only\; if\; \}\backslash ,\; A\; \backslash cap\; f^\{-1\}(B)\; =\; \backslash varnothing$

(equal if $f$ is injectiveSee ) , $f^\{-1\}(S\; \backslash cap\; T)\; =\; f^\{-1\}(S)\; \backslash cap\; f^\{-1\}(T)$ , - , $f(A\; \backslash setminus\; B)\; \backslash supseteq\; f(A)\; \backslash setminus\; f(B)$

(equal if $f$ is injective) , $f^\{-1\}(S\; \backslash setminus\; T)\; =\; f^\{-1\}(S)\; \backslash setminus\; f^\{-1\}(T)$ , - , $f\backslash left(A\; \backslash triangle\; B\backslash right)\; \backslash supseteq\; f(A)\; \backslash triangle\; f(B)$

(equal if $f$ is injective) , $f^\{-1\}\backslash left(S\; \backslash triangle\; T\backslash right)\; =\; f^\{-1\}(S)\; \backslash triangle\; f^\{-1\}(T)$ , - The results relating images and preimages to the (

element
Element or elements may refer to:
Science
* Chemical element, a pure substance of one type of atom
* Heating element, a device that generates heat by electrical resistance
* Orbital elements, parameters required to identify a specific orbit of o ...

of a given subset $A$ 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
* ...

produces a set, called the "image of $A$ under (or through) $f$". Similarly, the inverse image (or preimage) of a given subset $B$ of the codomain of $f,$ is the set of all elements of the domain that map to the members of $B.$
Image and inverse image may also be defined for general binary relations, not just functions.
Definition

The word "image" is used in three related ways. In these definitions, $f\; :\; X\; \backslash to\; Y$ is a function from theset
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...

$X$ to the set $Y.$
Image of an element

If $x$ is a member of $X,$ then the image of $x$ under $f,$ denoted $f(x),$ is thevalue
Value or values may refer to:
Ethics and social
* Value (ethics) wherein said concept may be construed as treating actions themselves as abstract objects, associating value to them
** Values (Western philosophy) expands the notion of value bey ...

of $f$ when applied to $x.$ $f(x)$ is alternatively known as the output of $f$ for argument $x.$
Given $y,$ the function $f$ is said to "" or "" if there exists some $x$ in the function's domain such that $f(x)\; =\; y.$
Similarly, given a set $S,$ $f$ is said to "" if there exists $x$ in the function's domain such that $f(x)\; \backslash in\; S.$
However, "" and "" means that $f(x)\; \backslash in\; S$ for point $x$ in $f$'s domain.
Image of a subset

Throughout, let $f\; :\; X\; \backslash to\; Y$ be a function. The under $f$ of a subset $A$ of $X$ is the set of all $f(a)$ for $a\backslash in\; A.$ It is denoted by $f;\; href="/html/ALL/s/.html"\; ;"title="">$ or by $f(A),$ when there is no risk of confusion. Usingset-builder notation
In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by enumerating its elements, or stating the properties that its members must satisfy.
Definin ...

, this definition can be written as
$$f;\; href="/html/ALL/s/.html"\; ;"title="">$$
This induces a function $f;\; href="/html/ALL/s/,\backslash cdot\backslash ,.html"\; ;"title=",\backslash cdot\backslash ,">,\backslash cdot\backslash ,$ where $\backslash mathcal\; P(S)$ denotes the power set of a set $S;$ that is the set of all subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...

s of $S.$ See below for more.
Image of a function

The ''image'' of a function is the image of its entiredomain
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
* ...

, also known as the range
Range may refer to:
Geography
* Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra)
** Mountain range, a group of mountains bordered by lowlands
* Range, a term used to i ...

of the function. This last usage should be avoided because the word "range" is also commonly used to mean the codomain of $f.$
Generalization to binary relations

If $R$ is an arbitrarybinary 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 sets and is a new set of ordered pairs consisting of elements in and i ...

on $X\; \backslash times\; Y,$ then the set $\backslash $ is called the image, or the range, of $R.$ Dually, the set $\backslash $ is called the domain of $R.$
Inverse image

Let $f$ be a function from $X$ to $Y.$ The preimage or inverse image of a set $B\; \backslash subseteq\; Y$ under $f,$ denoted by $f^;\; href="/html/ALL/s/.html"\; ;"title="">$ is the subset of $X$ defined by $$f^;\; href="/html/ALL/s/B\_.html"\; ;"title="B\; ">B$$ Other notations include $f^(B)$ and $f^(B).$ The inverse image of asingleton set
In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0.
Properties
Within the framework of Zermeloâ€“Fraenkel set theory, the ...

, denoted by $f^;\; href="/html/ALL/s/.html"\; ;"title="">$fiber
Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorporate ...

or fiber over $y$ or the level set
In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is:
: L_c(f) = \left\~,
When the number of independent variables is two, a level set is call ...

of $y.$ The set of all the fibers over the elements of $Y$ is a family of sets indexed by $Y.$
For example, for the function $f(x)\; =\; x^2,$ the inverse image of $\backslash $ would be $\backslash .$ Again, if there is no risk of confusion, $f^;\; href="/html/ALL/s/.html"\; ;"title="">$Notation for image and inverse image

The traditional notations used in the previous section do not distinguish the original function $f\; :\; X\; \backslash to\; Y$ from the image-of-sets function $f\; :\; \backslash mathcal(X)\; \backslash to\; \backslash mathcal(Y)$; 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

* $f^\backslash rightarrow\; :\; \backslash mathcal(X)\; \backslash to\; \backslash mathcal(Y)$ with $f^\backslash rightarrow(A)\; =\; \backslash $ * $f^\backslash leftarrow\; :\; \backslash mathcal(Y)\; \backslash to\; \backslash mathcal(X)$ with $f^\backslash leftarrow(B)\; =\; \backslash $Star notation

* $f\_\backslash star\; :\; \backslash mathcal(X)\; \backslash to\; \backslash mathcal(Y)$ instead of $f^\backslash rightarrow$ * $f^\backslash star\; :\; \backslash mathcal(Y)\; \backslash to\; \backslash mathcal(X)$ instead of $f^\backslash leftarrow$Other terminology

* An alternative notation for $f;\; href="/html/ALL/s/.html"\; ;"title="">$set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...

is $f\backslash ,\text{'}\text{'}A.$
* Some texts refer to the image of $f$ as the range of $f,$ but this usage should be avoided because the word "range" is also commonly used to mean the codomain of $f.$
Examples

# $f\; :\; \backslash \; \backslash to\; \backslash $ defined by $\backslash left\backslash \{\backslash begin\{matrix\}\; 1\; \backslash mapsto\; a,\; \backslash \backslash \; 2\; \backslash mapsto\; a,\; \backslash \backslash \; 3\; \backslash mapsto\; c.\; \backslash end\{matrix\}\backslash right.$ The ''image'' of the set $\backslash \{\; 2,\; 3\; \backslash \}$ under $f$ is $f(\backslash \{\; 2,\; 3\; \backslash \})\; =\; \backslash \{\; a,\; c\; \backslash \}.$ The ''image'' of the function $f$ is $\backslash \{\; a,\; c\; \backslash \}.$ The ''preimage'' of $a$ is $f^\{-1\}(\backslash \{\; a\; \backslash \})\; =\; \backslash \{\; 1,\; 2\; \backslash \}.$ The ''preimage'' of $\backslash \{\; a,\; b\; \backslash \}$ is also $f^\{-1\}(\backslash \{\; a,\; b\; \backslash \})\; =\; \backslash \{\; 1,\; 2\; \backslash \}.$ The ''preimage'' of $\backslash \{\; b,\; d\; \backslash \}$ under $f$ is the empty set $\backslash \{\; \backslash \; \backslash \}\; =\; \backslash emptyset.$ # $f\; :\; \backslash R\; \backslash to\; \backslash R$ defined by $f(x)\; =\; x^2.$ The ''image'' of $\backslash \{\; -2,\; 3\; \backslash \}$ under $f$ is $f(\backslash \{\; -2,\; 3\; \backslash \})\; =\; \backslash \{\; 4,\; 9\; \backslash \},$ and the ''image'' of $f$ is $\backslash R^+$ (the set of all positive real numbers and zero). The ''preimage'' of $\backslash \{\; 4,\; 9\; \backslash \}$ under $f$ is $f^\{-1\}(\backslash \{\; 4,\; 9\; \backslash \})\; =\; \backslash \{\; -3,\; -2,\; 2,\; 3\; \backslash \}.$ The ''preimage'' of set $N\; =\; \backslash \{\; n\; \backslash in\; \backslash R\; :\; n\; <\; 0\; \backslash \}$ under $f$ is the empty set, because the negative numbers do not have square roots in the set of reals. # $f\; :\; \backslash R^2\; \backslash to\; \backslash R$ defined by $f(x,\; y)\; =\; x^2\; +\; y^2.$ The ''fibers'' $f^\{-1\}(\backslash \{\; a\; \backslash \})$ areconcentric circles
In geometry, two or more objects are said to be concentric, coaxal, or coaxial when they share the same center or axis. Circles, regular polygons and regular polyhedra, and spheres may be concentric to one another (sharing the same center po ...

about the origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* ''The Origin'' (Buffy comic), a 1999 ''Buffy the Vampire Sl ...

, the origin itself, and the empty set (respectively), depending on whether $a\; >\; 0,\; \backslash \; a\; =\; 0,\; \backslash text\{\; or\; \}\; \backslash \; a\; <\; 0$ (respectively). (If $a\; \backslash ge\; 0,$ then the ''fiber'' $f^\{-1\}(\backslash \{\; a\; \backslash \})$ is the set of all $(x,\; y)\; \backslash in\; \backslash R^2$ satisfying the equation $x^2\; +\; y^2\; =\; a,$ that is, the origin-centered circle with radius $\backslash sqrt\{a\}.$)
# If $M$ is a manifold and $\backslash pi\; :\; TM\; \backslash to\; M$ is the canonical projection from the tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and ...

$TM$ to $M,$ then the ''fibers'' of $\backslash pi$ are the tangent spaces $T\_x(M)\; \backslash text\{\; for\; \}\; x\; \backslash in\; M.$ This is also an example of a fiber bundle.
# A quotient group is a homomorphic ''image''.
Properties

{, class=wikitable style="float:right;" , + ! Counter-examples based on the real numbers $\backslash R,$$f\; :\; \backslash R\; \backslash to\; \backslash R$ defined by $x\; \backslash mapsto\; x^2,$

showing that equality generally need

not hold for some laws: , - , , - , , - ,

General

For every function $f\; :\; X\; \backslash to\; Y$ and all subsets $A\; \backslash subseteq\; X$ and $B\; \backslash subseteq\; Y,$ the following properties hold: {, class="wikitable" , - ! Image ! Preimage , - , $f(X)\; \backslash subseteq\; Y$ , $f^\{-1\}(Y)\; =\; X$ , - , $f\backslash left(f^\{-1\}(Y)\backslash right)\; =\; f(X)$ , $f^\{-1\}(f(X))\; =\; X$ , - , $f\backslash left(f^\{-1\}(B)\backslash right)\; \backslash subseteq\; B$(equal if $B\; \backslash subseteq\; f(X);$ for instance, if $f$ is surjective)See See , $f^\{-1\}(f(A))\; \backslash supseteq\; A$

(equal if $f$ is injective) , - , $f(f^\{-1\}(B))\; =\; B\; \backslash cap\; f(X)$ , $\backslash left(f\; \backslash vert\_A\backslash right)^\{-1\}(B)\; =\; A\; \backslash cap\; f^\{-1\}(B)$ , - , $f\backslash left(f^\{-1\}(f(A))\backslash right)\; =\; f(A)$ , $f^\{-1\}\backslash left(f\backslash left(f^\{-1\}(B)\backslash right)\backslash right)\; =\; f^\{-1\}(B)$ , - , $f(A)\; =\; \backslash varnothing\; \backslash ,\backslash text\{\; if\; and\; only\; if\; \}\backslash ,\; A\; =\; \backslash varnothing$ , $f^\{-1\}(B)\; =\; \backslash varnothing\; \backslash ,\backslash text\{\; if\; and\; only\; if\; \}\backslash ,\; B\; \backslash subseteq\; Y\; \backslash setminus\; f(X)$ , - , $f(A)\; \backslash supseteq\; B\; \backslash ,\backslash text\{\; if\; and\; only\; if\; \}\; \backslash text\{\; there\; exists\; \}\; C\; \backslash subseteq\; A\; \backslash text\{\; such\; that\; \}\; f(C)\; =\; B$ , $f^\{-1\}(B)\; \backslash supseteq\; A\; \backslash ,\backslash text\{\; if\; and\; only\; if\; \}\backslash ,\; f(A)\; \backslash subseteq\; B$ , - , $f(A)\; \backslash supseteq\; f(X\; \backslash setminus\; A)\; \backslash ,\backslash text\{\; if\; and\; only\; if\; \}\backslash ,\; f(A)\; =\; f(X)$ , $f^\{-1\}(B)\; \backslash supseteq\; f^\{-1\}(Y\; \backslash setminus\; B)\; \backslash ,\backslash text\{\; if\; and\; only\; if\; \}\backslash ,\; f^\{-1\}(B)\; =\; X$ , - , $f(X\; \backslash setminus\; A)\; \backslash supseteq\; f(X)\; \backslash setminus\; f(A)$ , $f^\{-1\}(Y\; \backslash setminus\; B)\; =\; X\; \backslash setminus\; f^\{-1\}(B)$ , - , $f\backslash left(A\; \backslash cup\; f^\{-1\}(B)\backslash right)\; \backslash subseteq\; f(A)\; \backslash cup\; B$See p.388 of Lee, John M. (2010). Introduction to Topological Manifolds, 2nd Ed. , $f^\{-1\}(f(A)\; \backslash cup\; B)\; \backslash supseteq\; A\; \backslash cup\; f^\{-1\}(B)$ , - , $f\backslash left(A\; \backslash cap\; f^\{-1\}(B)\backslash right)\; =\; f(A)\; \backslash cap\; B$ , $f^\{-1\}(f(A)\; \backslash cap\; B)\; \backslash supseteq\; A\; \backslash cap\; f^\{-1\}(B)$ Also: * $f(A)\; \backslash cap\; B\; =\; \backslash varnothing\; \backslash ,\backslash text\{\; if\; and\; only\; if\; \}\backslash ,\; A\; \backslash cap\; f^\{-1\}(B)\; =\; \backslash varnothing$

Multiple functions

For functions $f\; :\; X\; \backslash to\; Y$ and $g\; :\; Y\; \backslash to\; Z$ with subsets $A\; \backslash subseteq\; X$ and $C\; \backslash subseteq\; Z,$ the following properties hold: * $(g\; \backslash circ\; f)(A)\; =\; g(f(A))$ * $(g\; \backslash circ\; f)^\{-1\}(C)\; =\; f^\{-1\}(g^\{-1\}(C))$Multiple subsets of domain or codomain

For function $f\; :\; X\; \backslash to\; Y$ and subsets $A,\; B\; \backslash subseteq\; X$ and $S,\; T\; \backslash subseteq\; Y,$ the following properties hold: {, class="wikitable" , - ! Image ! Preimage , - , $A\; \backslash subseteq\; B\; \backslash ,\backslash text\{\; implies\; \}\backslash ,\; f(A)\; \backslash subseteq\; f(B)$ , $S\; \backslash subseteq\; T\; \backslash ,\backslash text\{\; implies\; \}\backslash ,\; f^\{-1\}(S)\; \backslash subseteq\; f^\{-1\}(T)$ , - , $f(A\; \backslash cup\; B)\; =\; f(A)\; \backslash cup\; f(B)$ , $f^\{-1\}(S\; \backslash cup\; T)\; =\; f^\{-1\}(S)\; \backslash cup\; f^\{-1\}(T)$ , - , $f(A\; \backslash cap\; B)\; \backslash subseteq\; f(A)\; \backslash cap\; f(B)$(equal if $f$ is injectiveSee ) , $f^\{-1\}(S\; \backslash cap\; T)\; =\; f^\{-1\}(S)\; \backslash cap\; f^\{-1\}(T)$ , - , $f(A\; \backslash setminus\; B)\; \backslash supseteq\; f(A)\; \backslash setminus\; f(B)$

(equal if $f$ is injective) , $f^\{-1\}(S\; \backslash setminus\; T)\; =\; f^\{-1\}(S)\; \backslash setminus\; f^\{-1\}(T)$ , - , $f\backslash left(A\; \backslash triangle\; B\backslash right)\; \backslash supseteq\; f(A)\; \backslash triangle\; f(B)$

(equal if $f$ is injective) , $f^\{-1\}\backslash left(S\; \backslash triangle\; T\backslash right)\; =\; f^\{-1\}(S)\; \backslash triangle\; f^\{-1\}(T)$ , - The results relating images and preimages to the (

Boolean
Any kind of logic, function, expression, or theory based on the work of George Boole is considered Boolean.
Related to this, "Boolean" may refer to:
* Boolean data type, a form of data with only two possible values (usually "true" and "false" ...

) algebra of intersection and union work for any collection of subsets, not just for pairs of subsets:
* $f\backslash left(\backslash bigcup\_\{s\backslash in\; S\}A\_s\backslash right)\; =\; \backslash bigcup\_\{s\backslash in\; S\}\; f\backslash left(A\_s\backslash right)$
* $f\backslash left(\backslash bigcap\_\{s\backslash in\; S\}A\_s\backslash right)\; \backslash subseteq\; \backslash bigcap\_\{s\backslash in\; S\}\; f\backslash left(A\_s\backslash right)$
* $f^\{-1\}\backslash left(\backslash bigcup\_\{s\backslash in\; S\}B\_s\backslash right)\; =\; \backslash bigcup\_\{s\backslash in\; S\}\; f^\{-1\}\backslash left(B\_s\backslash right)$
* $f^\{-1\}\backslash left(\backslash bigcap\_\{s\backslash in\; S\}B\_s\backslash right)\; =\; \backslash bigcap\_\{s\backslash in\; S\}\; f^\{-1\}\backslash left(B\_s\backslash right)$
(Here, $S$ can be infinite, even uncountably infinite
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...

.)
With respect to the algebra of subsets described above, the inverse image function is a lattice homomorphism, while the image function is only a semilattice
In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has a ...

homomorphism (that is, it does not always preserve intersections).
See also

* * * * *Notes

References

* * . * * * * {{PlanetMath attribution, id=3276, title=Fibre Basic concepts in set theory Isomorphism theorems