Image (mathematics)

Inverse image

Let ${\displaystyle f}$ be a function from ${\displaystyle X}$ to ${\displaystyle Y.}$ The preimage or inverse image of a set ${\displaystyle B\subseteq Y}$ under ${\displaystyle f,}$ denoted by ${\displaystyle f^{-1}[B],}$ is the subset of ${\displaystyle X}$ defined by $${\displaystyle f^{-1}[B]=\{x\in X:f(x)\in B\}.}$$

Other notations include ${\displaystyle f^{-1}(B)}$ and ${\displaystyle f^{-}(B).}$ The inverse image of a singleton set, denoted by ${\displaystyle f^{-1}[\{y\}]}$ or by ${\displaystyle f^{-1}(y),}$ is also called the fiber (or fibre) over ${\displaystyle y}$, or the level set of ${\displaystyle y.}$ The set of all the fibers over the elements of ${\displaystyle Y}$ is a family of sets indexed by ${\displaystyle Y.}$

For example, for the function ${\displaystyle f(x)=x^2,}$ the inverse image of ${\displaystyle \{4\}}$ would be ${\displaystyle \{-2,2\}.}$ Again, if there is no risk of confusion, ${\displaystyle f^{-1}[B]}$ can be denoted by ${\displaystyle f^{-1}(B),}$ and ${\displaystyle f^{-1}}$ can also be thought of as a function from the power set of ${\displaystyle Y}$ to the power set of ${\displaystyle X.}$ The notation ${\displaystyle f^{-1}}$ should not be confused with that for inverse function, although it coincides with the usual one for bijections in that the inverse image of ${\displaystyle B}$ under ${\displaystyle f}$ is the image of ${\displaystyle B}$ under ${\displaystyle f^{-1}.}$

Notation for image and inverse image

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

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

Star notation

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

Other terminology

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

Examples

1. ${\displaystyle f:\{1,2,3\}\to \{a,b,c,d\}}$ defined by ${\displaystyle \{\begin{array}{c}1\mapsto a,\\ 2\mapsto a,\\ 3\mapsto c.\end{array}}$ The image of the set ${\displaystyle \{2,3\}}$ under ${\displaystyle f}$ is ${\displaystyle f(\{2,3\})=\{a,c\}.}$ The image of the function ${\displaystyle f}$ is ${\displaystyle \{a,c\}.}$ The preimage of ${\displaystyle a}$ is ${\displaystyle f^{-1}(\{a\})=\{1,2\}.}$ The preimage of ${\displaystyle \{a,b\}}$ is also ${\displaystyle f^{-1}(\{a,b\})=\{1,2\}.}$ The preimage of ${\displaystyle \{b,d\}}$ under ${\displaystyle f}$ is the empty set ${\displaystyle \{\}=\mathrm{\varnothing }.}$
2. ${\displaystyle f:\mathbb{R}\to \mathbb{R}}$ defined by ${\displaystyle f(x)=x^2.}$ The image of ${\displaystyle \{-2,3\}}$ under ${\displaystyle f}$ is ${\displaystyle f(\{-2,3\})=\{4,9\},}$ and the image of ${\displaystyle f}$ is ${\displaystyle {\mathbb{R}}^{+}}$ (the set of all positive real numbers and zero). The preimage of ${\displaystyle \{4,9\}}$ under ${\displaystyle f}$ is ${\displaystyle f^{-1}(\{4,9\})=\{-3,-2,2,3\}.}$ The preimage of set ${\displaystyle N=\{n\in \mathbb{R}:n<0\}}$ under ${\displaystyle f}$ is the empty set, because the negative numbers do not have square roots in the set of reals.
3. ${\displaystyle f:{\mathbb{R}}^2\to \mathbb{R}}$ defined by ${\displaystyle f(x,y)=x^2+y^2.}$ The fibers ${\displaystyle f^{-1}(\{a\})}$ are concentric circles about the origin, the origin itself, and the empty set (respectively), depending on whether ${\displaystyle a>0,a=0,\text{ or }a<0}$ (respectively). (If ${\displaystyle a\ge 0,}$ then the fiber ${\displaystyle f^{-1}(\{a\})}$ is the set of all ${\displaystyle (x,y)\in {\mathbb{R}}^2}$ satisfying the equation ${\displaystyle x^2+y^2=a,}$ that is, the origin-centered circle with radius ${\displaystyle \sqrt{a}.}$)
4. If ${\displaystyle M}$ is a manifold and ${\displaystyle \pi :TM\to M}$ is the canonical projection from the tangent bundle ${\displaystyle TM}$ to ${\displaystyle M,}$ then the fibers of ${\displaystyle \pi }$ are the tangent spaces ${\displaystyle T_x(M)\text{ for }x\in M.}$ This is also an example of a fiber bundle.
5. A quotient group is a homomorphic image.

Properties

General

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

Also:

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

Multiple functions

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:

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

Multiple subsets of domain or codomain

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

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).