image (functions)

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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 type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
is the set of all output values it may produce. More generally, evaluating a given function $f$ 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 $A$ of its
domain Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
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 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 ...

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 relation#Operations on binary relations, 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 Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
from the Set (mathematics), set $X$ 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 (mathematics), value 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 $f\left[A\right],$ or by $f\left(A\right),$ when there is no risk of confusion. Using set-builder notation, this definition can be written as $f[A] = \.$ This induces a function $f\left[\,\cdot\,\right] : \wp\left(X\right) \to \wp\left(Y\right),$ where $\wp\left(S\right)$ denotes the power set of a set $S;$ that is the set of all subsets 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 In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
, also known as the Range of a function, range of the function. This usage should be avoided because the word "range" is also commonly used to mean the
codomain 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 ...

of $f.$

## Generalization to binary relations

If $R$ is an arbitrary binary relation 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 $f^\left[B\right],$ is the subset of $X$ defined by $f^[ B ] = \.$ Other notations include $f^\left(B\right)$ and $f^\left(B\right).$ The inverse image of a Singleton (mathematics), singleton set, denoted by $f^\left[\\right]$ or by $f^\left[y\right],$ is also called the Fiber (mathematics), fiber or fiber over $y$ or the level set 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, $f^\left[B\right]$ can be denoted by $f^\left(B\right),$ and $f^$ can also be thought of as a function from the power set of $Y$ to the power set of $X.$ The notation $f^$ should not be confused with that for inverse function, although it coincides with the usual one for bijections in that the inverse image of $B$ under $f$ is the image of $B$ under $f^.$

# Notation for image and inverse image

The traditional notations used in the previous section may be confusing, because it does 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 it does not distinguish the inverse function (assuming one exists) from the inverse image function (which again relates the powersets). 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 $f\left[A\right]$ used in mathematical logic and set theory is $f\,\text{'}\text{'}A.$M. Randall Holmes
Inhomogeneity of the urelements in the usual models of NFU
December 29, 2005, on: Semantic Scholar, p. 2
* 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 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 ...

of $f.$

# Examples

# $f : \ \to \$ defined by $f\left(x\right) = \left\\left\{\begin\left\{matrix\right\} a, & \mbox\left\{if \right\}x=1 \\ a, & \mbox\left\{if \right\}x=2 \\ c, & \mbox\left\{if \right\}x=3. \end\left\{matrix\right\}\right.$ 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 $f^\left\{-1\right\}\left(\\left\{ 1, 2 \\right\}\right) = \\left\{ 1, 2 \\right\}.$ The preimage of $\\left\{ b, d \\right\},$ is the empty set $\\left\{ \, \\right\} = \varnothing.$ # $f : \R \to \R$ defined by $f\left(x\right) = x^2.$ The ''image'' of $\\left\{ -2, 3 \\right\}$ under $f$ is $f^\left\{-1\right\}\left(\\left\{ -2, 3 \\right\}\right) = \\left\{ 4, 9 \\right\},$ 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 Fiber (mathematics), ''fiber'' $f^\left\{-1\right\}\left(\\left\{ a \\right\}\right)$ are concentric circles about the Origin (mathematics), origin, the origin itself, and the empty set, depending on whether $a > 0, a = 0, \text\left\{ or \right\} a < 0,$ respectively. (if $a > 0,$ 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 of the origin-concentric ring $x^2 + y^2 = a.$) # If $M$ is a manifold and $\pi : TM \to M$ is the canonical Projection (mathematics), projection from the tangent bundle $TM$ 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 algebra (structure), Boolean) algebra of Intersection (set theory), intersection and Union (set theory), 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.) 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 homomorphism (that is, it does not always preserve intersections).