Image (mathematics)

If ${\displaystyle x}$ is a member of ${\displaystyle X,}$ then the image of ${\displaystyle x}$ under ${\displaystyle f,}$ denoted ${\displaystyle f(x),}$ is the value of ${\displaystyle f}$ when applied to ${\displaystyle x.}$ ${\displaystyle f(x)}$ is alternatively known as the output of ${\displaystyle f}$ for argument ${\displaystyle x.}$

Given ${\displaystyle y,}$ the function ${\displaystyle f}$ is said to "take the value ${\displaystyle y}$" or "take ${\displaystyle y}$ as a value" if there exists some ${\displaystyle x}$ in the function's domain such that ${\displaystyle f(x)=y.}$ Similarly, given a set ${\displaystyle S,}$ ${\displaystyle f}$ is said to "take a value in ${\displaystyle S}$" if there exists some ${\displaystyle x}$ in the function's domain such that ${\displaystyle f(x)\in S.}$ However, "${\displaystyle f}$ takes [all] values in ${\displaystyle S}$" and "${\displaystyle f}$ is valued in ${\displaystyle S}$" means that ${\displaystyle f(x)\in S}$ for every point ${\displaystyle x}$ in ${\displaystyle f}$'s domain.

Throughout, let ${\displaystyle f:X\to Y}$ be a function. The image under ${\displaystyle f}$ of a subset ${\displaystyle A}$ of ${\displaystyle X}$ is the set of all ${\displaystyle f(a)}$ for ${\displaystyle a\in A.}$ It is denoted by ${\displaystyle f[A],}$ or by ${\displaystyle f(A),}$ when there is no risk of confusion. Using set-builder notation, this definition can be written as[1][2]

$${\displaystyle f[A]=\{f(a):a\in A\}.}$$

This induces a function ${\displaystyle f[\,\cdot \,]:{\mathcal {P}}(X)\to {\mathcal {P}}(Y),}$ where ${\displaystyle {\mathcal {P}}(S)}$ denotes the power set of a set ${\displaystyle S;}$ that is the set of all subsets of ${\displaystyle S.}$ See § Notation below for more.

The image of a function is the image of its entire domain, also known as the range of the function.[3] This last usage should be avoided because the word "range" is also commonly used to mean the codomain of ${\displaystyle f.}$

If ${\displaystyle R}$ is an arbitrary binary relation on ${\displaystyle X\times Y,}$ then the set ${\displaystyle \{y\in Y:xRy{\text{ for some }}x\in X\}}$ is called the image, or the range, of ${\displaystyle R.}$ Dually, the set ${\displaystyle \{x\in X:xRy{\text{ for some }}y\in Y\}}$ is called the domain of ${\displaystyle R.}$

• ${\displaystyle f^{\rightarrow }:{\mathcal {P}}(X)\to {\mathcal {P}}(Y)}$ with ${\displaystyle f^{\rightarrow }(A)=\{f(a)\;|\;a\in A\}}$
• ${\displaystyle f^{\leftarrow }:{\mathcal {P}}(Y)\to {\mathcal {P}}(X)}$ with ${\displaystyle f^{\leftarrow }(B)=\{a\in X\;|\;f(a)\in B\}}$

• ${\displaystyle f_{\star }:{\mathcal {P}}(X)\to {\mathcal {P}}(Y)}$ instead of ${\displaystyle f^{\rightarrow }}$
• ${\displaystyle f^{\star }:{\mathcal {P}}(Y)\to {\mathcal {P}}(X)}$ instead of ${\displaystyle f^{\leftarrow }}$

• An alternative notation for ${\displaystyle f[A]}$ used in mathematical logic and set theory is ${\displaystyle f\,''A.}$[6][7]
• Some texts refer to the image of ${\displaystyle f}$ as the range of ${\displaystyle f,}$[8] but this usage should be avoided because the word "range" is also commonly used to mean the codomain of ${\displaystyle f.}$

For every function ${\displaystyle f:X\to Y}$ and all subsets ${\displaystyle A\subseteq X}$ and ${\displaystyle B\subseteq Y,}$ the following properties hold:

Image	Preimage
${\displaystyle f(X)\subseteq Y}$	${\displaystyle f^{-1}(Y)=X}$
${\displaystyle f\left(f^{-1}(Y)\right)=f(X)}$	${\displaystyle f^{-1}(f(X))=X}$
${\displaystyle f\left(f^{-1}(B)\right)\subseteq B}$ (equal if ${\displaystyle B\subseteq f(X);}$ for instance, if ${\displaystyle f}$ is surjective)[9][10]	${\displaystyle f^{-1}(f(A))\supseteq A}$ (equal if ${\displaystyle f}$ is injective)[9][10]
${\displaystyle f(f^{-1}(B))=B\cap f(X)}$	${\displaystyle \left(f\vert _{A}\right)^{-1}(B)=A\cap f^{-1}(B)}$
${\displaystyle f\left(f^{-1}(f(A))\right)=f(A)}$	${\displaystyle f^{-1}\left(f\left(f^{-1}(B)\right)\right)=f^{-1}(B)}$
${\displaystyle f(A)=\varnothing \,{\text{ if and only if }}\,A=\varnothing }$	${\displaystyle f^{-1}(B)=\varnothing \,{\text{ if and only if }}\,B\subseteq Y\setminus f(X)}$
${\displaystyle f(A)\supseteq B\,{\text{ if and only if }}{\text{ there exists }}C\subseteq A{\text{ such that }}f(C)=B}$	${\displaystyle f^{-1}(B)\supseteq A\,{\text{ if and only if }}\,f(A)\subseteq B}$
${\displaystyle f(A)\supseteq f(X\setminus A)\,{\text{ if and only if }}\,f(A)=f(X)}$	${\displaystyle f^{-1}(B)\supseteq f^{-1}(Y\setminus B)\,{\text{ if and only if }}\,f^{-1}(B)=X}$
${\displaystyle f(X\setminus A)\supseteq f(X)\setminus f(A)}$	${\displaystyle f^{-1}(Y\setminus B)=X\setminus f^{-1}(B)}$[9]
${\displaystyle f\left(A\cup f^{-1}(B)\right)\subseteq f(A)\cup B}$[11]	${\displaystyle f^{-1}(f(A)\cup B)\supseteq A\cup f^{-1}(B)}$[11]
${\displaystyle f\left(A\cap f^{-1}(B)\right)=f(A)\cap B}$[11]	${\displaystyle f^{-1}(f(A)\cap B)\supseteq A\cap f^{-1}(B)}$[11]

Also:

• ${\displaystyle f(A)\cap B=\varnothing \,{\text{ if and only if }}\,A\cap f^{-1}(B)=\varnothing }$

For functions ${\displaystyle f:X\to Y}$ and ${\displaystyle g:Y\to Z}$ with subsets ${\displaystyle A\subseteq X}$ and ${\displaystyle C\subseteq Z,}$ the following properties hold:

• ${\displaystyle (g\circ f)(A)=g(f(A))}$
• ${\displaystyle (g\circ f)^{-1}(C)=f^{-1}(g^{-1}(C))}$

For function ${\displaystyle f:X\to Y}$ and subsets ${\displaystyle A,B\subseteq X}$ and ${\displaystyle S,T\subseteq Y,}$ the following properties hold:

Image	Preimage
${\displaystyle A\subseteq B\,{\text{ implies }}\,f(A)\subseteq f(B)}$	${\displaystyle S\subseteq T\,{\text{ implies }}\,f^{-1}(S)\subseteq f^{-1}(T)}$
${\displaystyle f(A\cup B)=f(A)\cup f(B)}$[11][12]	${\displaystyle f^{-1}(S\cup T)=f^{-1}(S)\cup f^{-1}(T)}$
${\displaystyle f(A\cap B)\subseteq f(A)\cap f(B)}$[11][12] (equal if ${\displaystyle f}$ is injective[13])	${\displaystyle f^{-1}(S\cap T)=f^{-1}(S)\cap f^{-1}(T)}$
${\displaystyle f(A\setminus B)\supseteq f(A)\setminus f(B)}$[11] (equal if ${\displaystyle f}$ is injective[13])	${\displaystyle f^{-1}(S\setminus T)=f^{-1}(S)\setminus f^{-1}(T)}$[11]
${\displaystyle f\left(A\triangle B\right)\supseteq f(A)\triangle f(B)}$ (equal if ${\displaystyle f}$ is injective)	${\displaystyle f^{-1}\left(S\triangle T\right)=f^{-1}(S)\triangle f^{-1}(T)}$

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:

• ${\displaystyle f\left(\bigcup _{s\in S}A_{s}\right)=\bigcup _{s\in S}f\left(A_{s}\right)}$
• ${\displaystyle f\left(\bigcap _{s\in S}A_{s}\right)\subseteq \bigcap _{s\in S}f\left(A_{s}\right)}$
• ${\displaystyle f^{-1}\left(\bigcup _{s\in S}B_{s}\right)=\bigcup _{s\in S}f^{-1}\left(B_{s}\right)}$
• ${\displaystyle f^{-1}\left(\bigcap _{s\in S}B_{s}\right)=\bigcap _{s\in S}f^{-1}\left(B_{s}\right)}$

(Here, ${\displaystyle S}$ 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).