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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
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, $f : X \to Y$ is a function from the
set 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 $f\left(x\right),$ is the
value 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 $x.$ $f\left(x\right)$ 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 $f\left(x\right) = y.$ Similarly, given a set $S,$ $f$ is said to "" if there exists $x$ in the function's domain such that $f\left(x\right) \in S.$ However, "" and "" means that $f\left(x\right) \in S$ for point $x$ in $f$'s domain.

## Image of a subset

Throughout, let $f : X \to Y$ be a function. The under $f$ of a subset $A$ of $X$ is the set of all $f\left(a\right)$ for $a\in A.$ It is denoted by or by $f\left(A\right),$ when there is no risk of confusion. Using
set-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 This induces a function where $\mathcal P\left(S\right)$ 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 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 * ...
, 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 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 sets and is a new set of ordered pairs consisting of elements in and i ...
on $X \times Y,$ then the set $\$ is called the image, or the range, of $R.$ Dually, the set $\$ 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 $B \subseteq Y$ under $f,$ denoted by is the subset of $X$ defined by Other notations include $f^\left(B\right)$ and $f^\left(B\right).$ The inverse image of a
singleton 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 Zermelo–Fraenkel set theory, the ...
, denoted by
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 $f\left(x\right) = x^2,$ the inverse image of $\$ would be $\.$ Again, if there is no risk of confusion,

# Notation for image and inverse image

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

* $f^\rightarrow : \mathcal\left(X\right) \to \mathcal\left(Y\right)$ with $f^\rightarrow\left(A\right) = \$ * $f^\leftarrow : \mathcal\left(Y\right) \to \mathcal\left(X\right)$ with $f^\leftarrow\left(B\right) = \$

## Star notation

* $f_\star : \mathcal\left(X\right) \to \mathcal\left(Y\right)$ instead of $f^\rightarrow$ * $f^\star : \mathcal\left(Y\right) \to \mathcal\left(X\right)$ instead of $f^\leftarrow$

## Other terminology

* An alternative notation for
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 $f\,\text{'}\text{'}A.$ * 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 : \ \to \$ defined by $\left\\left\{\begin\left\{matrix\right\} 1 \mapsto a, \\ 2 \mapsto a, \\ 3 \mapsto c. \end\left\{matrix\right\}\right.$ The ''image'' of the set $\\left\{ 2, 3 \\right\}$ under $f$ is $f\left(\\left\{ 2, 3 \\right\}\right) = \\left\{ a, c \\right\}.$ The ''image'' of the function $f$ is $\\left\{ a, c \\right\}.$ The ''preimage'' of $a$ is $f^\left\{-1\right\}\left(\\left\{ a \\right\}\right) = \\left\{ 1, 2 \\right\}.$ The ''preimage'' of $\\left\{ a, b \\right\}$ is also $f^\left\{-1\right\}\left(\\left\{ a, b \\right\}\right) = \\left\{ 1, 2 \\right\}.$ The ''preimage'' of $\\left\{ b, d \\right\}$ under $f$ is the empty set $\\left\{ \ \\right\} = \emptyset.$ # $f : \R \to \R$ defined by $f\left(x\right) = x^2.$ The ''image'' of $\\left\{ -2, 3 \\right\}$ under $f$ is $f\left(\\left\{ -2, 3 \\right\}\right) = \\left\{ 4, 9 \\right\},$ and the ''image'' of $f$ is $\R^+$ (the set of all positive real numbers and zero). The ''preimage'' of $\\left\{ 4, 9 \\right\}$ under $f$ is $f^\left\{-1\right\}\left(\\left\{ 4, 9 \\right\}\right) = \\left\{ -3, -2, 2, 3 \\right\}.$ The ''preimage'' of set $N = \\left\{ n \in \R : n < 0 \\right\}$ under $f$ is the empty set, because the negative numbers do not have square roots in the set of reals. # $f : \R^2 \to \R$ defined by $f\left(x, y\right) = x^2 + y^2.$ The ''fibers'' $f^\left\{-1\right\}\left(\\left\{ a \\right\}\right)$ are
concentric 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 ...
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 $a > 0, \ a = 0, \text\left\{ or \right\} \ a < 0$ (respectively). (If $a \ge 0,$ then the ''fiber'' $f^\left\{-1\right\}\left(\\left\{ a \\right\}\right)$ is the set of all $\left(x, y\right) \in \R^2$ satisfying the equation $x^2 + y^2 = a,$ that is, the origin-centered circle with radius $\sqrt\left\{a\right\}.$) # If $M$ is a manifold and $\pi : TM \to M$ 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 $\pi$ are the tangent spaces $T_x\left(M\right) \text\left\{ for \right\} x \in M.$ 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 $\R,$
$f : \R \to \R$ defined by $x \mapsto x^2,$
showing that equality generally need
not hold for some laws: , - , , - , , - ,

## General

For every function $f : X \to Y$ and all subsets $A \subseteq X$ and $B \subseteq Y,$ the following properties hold: {, class="wikitable" , - ! Image ! Preimage , - , $f\left(X\right) \subseteq Y$ , $f^\left\{-1\right\}\left(Y\right) = X$ , - , $f\left\left(f^\left\{-1\right\}\left(Y\right)\right\right) = f\left(X\right)$ , $f^\left\{-1\right\}\left(f\left(X\right)\right) = X$ , - , $f\left\left(f^\left\{-1\right\}\left(B\right)\right\right) \subseteq B$
(equal if $B \subseteq f\left(X\right);$ for instance, if $f$ is surjective)See See , $f^\left\{-1\right\}\left(f\left(A\right)\right) \supseteq A$
(equal if $f$ is injective) , - , $f\left(f^\left\{-1\right\}\left(B\right)\right) = B \cap f\left(X\right)$ , $\left\left(f \vert_A\right\right)^\left\{-1\right\}\left(B\right) = A \cap f^\left\{-1\right\}\left(B\right)$ , - , $f\left\left(f^\left\{-1\right\}\left(f\left(A\right)\right)\right\right) = f\left(A\right)$ , $f^\left\{-1\right\}\left\left(f\left\left(f^\left\{-1\right\}\left(B\right)\right\right)\right\right) = f^\left\{-1\right\}\left(B\right)$ , - , $f\left(A\right) = \varnothing \,\text\left\{ if and only if \right\}\, A = \varnothing$ , $f^\left\{-1\right\}\left(B\right) = \varnothing \,\text\left\{ if and only if \right\}\, B \subseteq Y \setminus f\left(X\right)$ , - , $f\left(A\right) \supseteq B \,\text\left\{ if and only if \right\} \text\left\{ there exists \right\} C \subseteq A \text\left\{ such that \right\} f\left(C\right) = B$ , $f^\left\{-1\right\}\left(B\right) \supseteq A \,\text\left\{ if and only if \right\}\, f\left(A\right) \subseteq B$ , - , $f\left(A\right) \supseteq f\left(X \setminus A\right) \,\text\left\{ if and only if \right\}\, f\left(A\right) = f\left(X\right)$ , $f^\left\{-1\right\}\left(B\right) \supseteq f^\left\{-1\right\}\left(Y \setminus B\right) \,\text\left\{ if and only if \right\}\, f^\left\{-1\right\}\left(B\right) = X$ , - , $f\left(X \setminus A\right) \supseteq f\left(X\right) \setminus f\left(A\right)$ , $f^\left\{-1\right\}\left(Y \setminus B\right) = X \setminus f^\left\{-1\right\}\left(B\right)$ , - , $f\left\left(A \cup f^\left\{-1\right\}\left(B\right)\right\right) \subseteq f\left(A\right) \cup B$See p.388 of Lee, John M. (2010). Introduction to Topological Manifolds, 2nd Ed. , $f^\left\{-1\right\}\left(f\left(A\right) \cup B\right) \supseteq A \cup f^\left\{-1\right\}\left(B\right)$ , - , $f\left\left(A \cap f^\left\{-1\right\}\left(B\right)\right\right) = f\left(A\right) \cap B$ , $f^\left\{-1\right\}\left(f\left(A\right) \cap B\right) \supseteq A \cap f^\left\{-1\right\}\left(B\right)$ Also: * $f\left(A\right) \cap B = \varnothing \,\text\left\{ if and only if \right\}\, A \cap f^\left\{-1\right\}\left(B\right) = \varnothing$

## Multiple functions

For functions $f : X \to Y$ and $g : Y \to Z$ with subsets $A \subseteq X$ and $C \subseteq Z,$ the following properties hold: * $\left(g \circ f\right)\left(A\right) = g\left(f\left(A\right)\right)$ * $\left(g \circ f\right)^\left\{-1\right\}\left(C\right) = f^\left\{-1\right\}\left(g^\left\{-1\right\}\left(C\right)\right)$

## Multiple subsets of domain or codomain

For function $f : X \to Y$ and subsets $A, B \subseteq X$ and $S, T \subseteq Y,$ the following properties hold: {, class="wikitable" , - ! Image ! Preimage , - , $A \subseteq B \,\text\left\{ implies \right\}\, f\left(A\right) \subseteq f\left(B\right)$ , $S \subseteq T \,\text\left\{ implies \right\}\, f^\left\{-1\right\}\left(S\right) \subseteq f^\left\{-1\right\}\left(T\right)$ , - , $f\left(A \cup B\right) = f\left(A\right) \cup f\left(B\right)$ , $f^\left\{-1\right\}\left(S \cup T\right) = f^\left\{-1\right\}\left(S\right) \cup f^\left\{-1\right\}\left(T\right)$ , - , $f\left(A \cap B\right) \subseteq f\left(A\right) \cap f\left(B\right)$
(equal if $f$ is injectiveSee ) , $f^\left\{-1\right\}\left(S \cap T\right) = f^\left\{-1\right\}\left(S\right) \cap f^\left\{-1\right\}\left(T\right)$ , - , $f\left(A \setminus B\right) \supseteq f\left(A\right) \setminus f\left(B\right)$
(equal if $f$ is injective) , $f^\left\{-1\right\}\left(S \setminus T\right) = f^\left\{-1\right\}\left(S\right) \setminus f^\left\{-1\right\}\left(T\right)$ , - , $f\left\left(A \triangle B\right\right) \supseteq f\left(A\right) \triangle f\left(B\right)$
(equal if $f$ is injective) , $f^\left\{-1\right\}\left\left(S \triangle T\right\right) = f^\left\{-1\right\}\left(S\right) \triangle f^\left\{-1\right\}\left(T\right)$ , - 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: * $f\left\left(\bigcup_\left\{s\in S\right\}A_s\right\right) = \bigcup_\left\{s\in S\right\} f\left\left(A_s\right\right)$ * $f\left\left(\bigcap_\left\{s\in S\right\}A_s\right\right) \subseteq \bigcap_\left\{s\in S\right\} f\left\left(A_s\right\right)$ * $f^\left\{-1\right\}\left\left(\bigcup_\left\{s\in S\right\}B_s\right\right) = \bigcup_\left\{s\in S\right\} f^\left\{-1\right\}\left\left(B_s\right\right)$ * $f^\left\{-1\right\}\left\left(\bigcap_\left\{s\in S\right\}B_s\right\right) = \bigcap_\left\{s\in S\right\} f^\left\{-1\right\}\left\left(B_s\right\right)$ (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).