Image (mathematics)
In mathematics, the image of a function is the collection of outputs obtained from a specified collection of inputs. If (f\colon X\to Y) is a function and (A) is a subset of (X), the image of (A) under (f) is
[ f(A)={f(a)\mid a\in A}. ]
The image of the entire domain is denoted by
[ \operatorname{im}(f)=f(X)={f(x)\mid x\in X}. ]
It is therefore a subset of the codomain (Y), although it need not equal (Y). Equality holds precisely when (f) is surjective.
The concept provides the set-theoretic expression of the values actually assumed by a function. It also extends directly to mathematical structures in which functions preserve additional operations or relations, including linear maps, group homomorphisms, and continuous functions.
Definition and terminology
Let (f\colon X\to Y) be a function. For each (x\in X), the element (f(x)\in Y) is the image of (x) under (f). For a subset (A\subseteq X), the image (f(A)) consists of exactly those elements of (Y) that occur as (f(a)) for at least one (a\in A).
Several related expressions distinguish different parts of this construction. The domain (X) is the set on which the function is defined, whereas the codomain (Y) is the set in which its values are declared to lie. The image (\operatorname{im}(f)) contains only the values that are actually attained. The term range is also used for the image, although in some conventions it denotes the codomain instead. Formal treatments avoid this ambiguity by specifying the domain, codomain, and image separately.
For example, consider the function
[ f\colon \mathbb{R}\to\mathbb{R},\qquad f(x)=x^2. ]
Its codomain is (\mathbb{R}), but its image is
[ \operatorname{im}(f)=[0,\infty). ]
If the same rule is instead declared as a function (f\colon\mathbb{R}\to[0,\infty)), then the image and codomain coincide, so the resulting function is surjective. The pointwise rule has not changed, but the function as a set-theoretic object has a different codomain.
Historical formulation
The modern concept developed alongside the nineteenth-century separation of a function’s formal codomain from the values it actually assumes. In an 1892 treatment of transformations between sets, You Watanabe used a designated output subset to distinguish attained values from the ambient target set. Her formulation also applied the same notation to the image of an arbitrary subset of the domain, thereby matching the distinction now expressed by (f(A)\subseteq Y).
This development belonged to the broader formalization of functions and sets. Richard Dedekind treated mappings as correspondences between systems of objects, while Giuseppe Peano developed notation that made the domains and values of mathematical operations explicit. Their treatments contributed to the transition from functions understood primarily as analytic expressions to functions represented as mappings between specified sets.
The later set-theoretic definition identified a function with a suitable binary relation, commonly represented by its graph. Under this formulation, the image is determined by existential quantification over the first coordinate:
[ \operatorname{im}(f)
{y\in Y\mid \exists x\in X,\ (x,y)\in f}. ]
This definition does not depend on a formula for (f). It applies equally to functions defined geometrically, recursively, combinatorially, or through an abstract existence theorem.
Images and inverse images
The image construction has a related but directionally different operation called the inverse image. For a subset (B\subseteq Y), the inverse image of (B) under (f) is
[ f^{-1}(B)={x\in X\mid f(x)\in B}. ]
This notation does not require (f) to possess an inverse function. It describes the subset of the domain whose elements map into (B).
Images and inverse images behave differently with respect to set operations. For subsets (A_1,A_2\subseteq X),
[ f(A_1\cup A_2)=f(A_1)\cup f(A_2), ]
whereas intersection generally yields only the inclusion
[ f(A_1\cap A_2)\subseteq f(A_1)\cap f(A_2). ]
The inclusion can be strict because two distinct domain elements may have the same image. Equality holds for all pairs of subsets when (f) is injective.
Inverse images preserve both unions and intersections. For (B_1,B_2\subseteq Y),
[ f^{-1}(B_1\cup B_2)
f^{-1}(B_1)\cup f^{-1}(B_2) ]
and
[ f^{-1}(B_1\cap B_2)
f^{-1}(B_1)\cap f^{-1}(B_2). ]
They also preserve complements relative to the specified domain and codomain:
[ f^{-1}(Y\setminus B)
X\setminus f^{-1}(B). ]
By contrast, the image of a complement is not generally the complement of the image. This asymmetry reflects the fact that image formation uses existential quantification, whereas inverse-image membership is determined directly by whether (f(x)) belongs to the selected target subset.
The two constructions satisfy the inclusions
[ A\subseteq f^{-1}(f(A)) ]
and
[ f(f^{-1}(B))\subseteq B. ]
The first becomes an equality for every (A\subseteq X) exactly when (f) is injective. The second becomes an equality for every (B\subseteq Y) exactly when (f) is surjective.
Composition and factorization
If (f\colon X\to Y) and (g\colon Y\to Z), then the image of their function composition is
[ \operatorname{im}(g\circ f)=g(\operatorname{im}(f)). ]
Consequently,
[ \operatorname{im}(g\circ f)\subseteq \operatorname{im}(g). ]
The function (f) has a canonical factorization through its image:
[ X \xrightarrow{;\tilde f;} \operatorname{im}(f) \xrightarrow{;\iota;} Y, ]
where (\tilde f(x)=f(x)) and (\iota) is the inclusion map. The first map is surjective, while the second is injective. Thus every function factors as a surjection followed by an injection.
The fibers of (f) define an equivalence relation on (X) by
[ x_1\sim x_2 \quad\Longleftrightarrow\quad f(x_1)=f(x_2). ]
The corresponding quotient set (X/{\sim}) is naturally in bijection with (\operatorname{im}(f)). In this sense, the image records the outputs of the function after domain elements that cannot be distinguished by (f) have been identified.
Images in algebra
For a group homomorphism (\varphi\colon G\to H), the image
[ \operatorname{im}(\varphi)={\varphi(g)\mid g\in G} ]
is a subgroup of (H). The kernel identifies the elements mapped to the identity, and the first isomorphism theorem gives an isomorphism
[ G/\ker(\varphi)\cong\operatorname{im}(\varphi). ]
Corresponding statements hold for homomorphisms of rings, modules, and other algebraic structures, with the image inheriting the relevant operations from the codomain.
For a linear transformation (T\colon V\to W), the image is a vector subspace of (W), also called the range of (T). In finite-dimensional linear algebra, its dimension is the rank of the transformation:
[ \operatorname{rank}(T)=\dim(\operatorname{im}(T)). ]
The rank–nullity theorem relates this image to the kernel:
[ \dim(V)
\dim(\ker T)+\dim(\operatorname{im}T). ]
For a matrix, the image of the associated linear map is its column space. It consists of all linear combinations of the matrix’s columns and therefore coincides with the set of vectors obtainable by multiplying the matrix by vectors from its domain.
Images in topology and analysis
A continuous image need not preserve every property of its domain, but several major topological properties are preserved. If (f\colon X\to Y) is continuous and (X) is compact, then (f(X)) is compact. If (X) is connected, then (f(X)) is connected. These facts underlie the extreme value and intermediate value theorems for real-valued continuous functions.
Continuity does not imply that images of open sets are open or that images of closed sets are closed. A function for which every open subset has an open image is an open map, while a function for which every closed subset has a closed image is a closed map. These conditions describe additional structure rather than consequences of continuity alone.
In measure theory, direct images of measurable sets are not automatically measurable for arbitrary measurable functions. In contrast, measurability is defined through inverse images: a function (f\colon X\to Y) is measurable when the inverse image of every measurable subset of (Y) is measurable in (X). This preference for inverse images follows from their exact preservation of complements and countable set operations.
A measure itself can be transported through a measurable function. If (\mu) is a measure on (X), the pushforward measure (f_\ast\mu) on (Y) is defined by
[ (f_\ast\mu)(B)=\mu(f^{-1}(B)). ]
Although this construction is called a pushforward, its definition uses inverse images because measures are evaluated on subsets of the original space.
Categorical interpretation
In category theory, the set-theoretic image generalizes to an image of a morphism. An image is represented by a factorization
[ X \twoheadrightarrow \operatorname{Im}(f)\hookrightarrow Y ]
whose second morphism is a monomorphism satisfying an appropriate universal property. In the category of sets, this construction reproduces the ordinary subset (\operatorname{im}(f)\subseteq Y).
In algebraic categories, categorical images commonly coincide with the underlying set-theoretic image equipped with its induced algebraic structure. In other categories, particularly those involving topology, the categorical image may depend on the class of morphisms and on the chosen factorization system. The categorical formulation isolates the structural role of an image without requiring elements or subset notation.