Functions:Forward Image

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

Image of a subset

The image of a subset $A\subseteq X$ under $f$, denoted $f[A]$, is the subset of $Y$ which can be defined using set-builder notation as follows:

$f[A]=\{f(x):x\in A\}$

When there is no risk of confusion, $f[A]$ is simply written as $f(A)$. This convention is a common one; the intended meaning must be inferred from the context. This makes $f[\,\cdot \,]$ a function whose domain is the power set of $X$ (the set of all subsets of $X$), and whose codomain is the power set of $Y$.

Image of a function

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

Generalization to binary relations

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

Examples

1. $f:\{1,2,3\}\to \{a,b,c,d\}$ defined by $f(x)=\left\{{\begin{matrix}a,&{\mbox{if }}x=1\\a,&{\mbox{if }}x=2\\c,&{\mbox{if }}x=3.\end{matrix}}\right.$

The image of the set $\{2,3\}$ under $f$ is $f(\{2,3\})=\{a,c\}.$ The image of the function $f$ is $\{a,c\}.$ The preimage of $a$ is $f^{-1}(\{a\})=\{1,2\}.$ The preimage of $\{a,b\}$ is also $f^{-1}(\{1,2\})=\{1,2\}.$ The preimage of $\{b,d\},$ is the empty set $\{\,\}=\varnothing .$

2. $f:\mathbb {R} \to \mathbb {R}$ defined by $f(x)=x^{2}.$

The image of $\{-2,3\}$ under $f$ is $f^{-1}(\{-2,3\})=\{4,9\},$ and the image of $f$ is $\mathbb {R} ^{+}$ (the set of all positive real numbers and zero). The preimage of $\{4,9\}$ under $f$ is $f^{-1}(\{4,9\})=\{-3,-2,2,3\}.$ The preimage of set $N=\{n\in \mathbb {R} :n<0\}$ under $f$ is the empty set, because the negative numbers do not have square roots in the set of reals.

3. $f:\mathbb {R} ^{2}\to \mathbb {R}$ defined by $f(x,y)=x^{2}+y^{2}.$

The fiber $f^{-1}(\{a\})$ are concentric circles]] about the origin, the origin itself, and the empty set, depending on whether $a>0,a=0,{\text{ or }}a<0,$ respectively. (if $a>0,$ then the fiber $f^{-1}(\{a\})$ is the set of all $(x,y)\in \mathbb {R} ^{2}$ satisfying the equation of the origin-concentric ring $x^{2}+y^{2}=a.$)

4. If $M$ is a manifold and $\pi :TM\to M$ is the canonical projection from the tangent bundle $TM$ to $M,$ then the fibers of $\pi$ are the tangent spaces $T_{x}(M){\text{ for }}x\in M.$ This is also an example of a fiber bundle.

5. A quotient group is a homomorphic image.

Properties

Counter-examples based on the real numbers $\mathbb {R} ,$ $f:\mathbb {R} \to \mathbb {R}$ defined by $x\mapsto x^{2},$ showing that equality generally need not hold for some laws:
Image showing non-equal sets: $f\left(A\cap B\right)\subsetneq f(A)\cap f(B).$ The sets $A=[-4,2]$ and $B=[-2,4]$ are shown in Template:Color immediately below the $x$-axis while their intersection $A_{3}=[-2,2]$ is shown in Template:Color.
$f\left(f^{-1}\left(B_{3}\right)\right)\subsetneq B_{3}.$
$f^{-1}\left(f\left(A_{4}\right)\right)\supsetneq A_{4}.$

General

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

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

Also:

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

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:

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

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:

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

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:

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

(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).

Resources

• Image (mathematics), Wikipedia

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See p.388 of Lee, John M. (2010). Introduction to Topological Manifolds, 2nd Ed.

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