Image of a Function · aiwiki.page

The image of a function is the set of all values produced by applying a function to every element of its domain. For a function $f:X\to Y$ , its image is denoted by $f(X)$ or $\operatorname{im}(f)$ and is a subset of the codomain $Y$ . More generally, the image of a subset $A\subseteq X$ consists of the outputs produced by inputs in $A$ . The image of the whole function is often called its range, although some authors use “range” to mean codomain instead. (jirka.org)

Definition and notation

For $f:X\to Y$ , the image of $A\subseteq X$ is defined by

f(A)=\{f(x):x\in A\} =\{y\in Y:\text{there exists }x\in A \text{ such that }f(x)=y\}.

This is also called the direct image of $A$ . Taking $A=X$ gives

\operatorname{im}(f)=f(X).

Thus membership in the image is an existence statement: an output belongs to $f(A)$ exactly when at least one input in $A$ produces it. (jirka.org)

The phrase “image of an element $x$ ” refers to the value $f(x)$ , whereas the image of the singleton set $\{x\}$ is the set $\{f(x)\}$ . The same notation $f$ is conventionally used both for evaluation on elements and for taking images of sets; the argument determines which operation is intended. (jirka.org)

Image, domain, and codomain

These three sets serve different roles:

The domain specifies the permitted inputs.
The codomain specifies the set in which outputs must lie.
The image consists of the outputs actually attained.

The domain and codomain are specified when declaring a function $f:X\to Y$ ; the image follows from its action on the domain. In particular, the codomain may contain elements that are never produced. (cs.cornell.edu)

For example, consider the real-valued function

f:\mathbb R\to\mathbb R,\qquad f(x)=x^2.

Its image is $[0,\infty)$ : every square is nonnegative, and every $y\geq0$ is attained by taking $x=\sqrt y$ . If the domain is restricted to $[-2,1]$ , the image becomes $[0,4]$ . If the domain is instead the integers, the image is the set of nonnegative integer squares. These are direct applications of the definition and illustrate why the formula alone does not determine the image without a specified domain. (jirka.org)

Changing the codomain while keeping the same inputs and outputs does not change the image, provided the new codomain still contains every output. However, it can change whether the function is surjective. (tildesites.geneseo.edu)

Surjectivity and invertibility

A surjective function is one whose image equals its codomain:

f\text{ is surjective}\quad\Longleftrightarrow\quad f(X)=Y.

An injective function never assigns the same output to two distinct inputs. A bijective function satisfies both conditions. (math.dartmouth.edu)

Every function can be regarded as a surjection onto its own image by replacing its codomain with $f(X)$ :

\widetilde f:X\to f(X),\qquad \widetilde f(x)=f(x).

If $f$ is also injective, this map is bijective and has an inverse function defined on $f(X)$ . These conclusions follow directly from the definitions: every element of the image has an input producing it, and injectivity makes that input unique. (math.dartmouth.edu)

Images and preimages

The preimage, or inverse image, of $B\subseteq Y$ is

f^{-1}(B)=\{x\in X:f(x)\in B\}.

An image moves from a set of inputs to the outputs they produce; a preimage moves from a set of outputs to all inputs producing those outputs. Preimage notation does not require an inverse function to exist. (jirka.org)

For the square function, direct calculation gives

f^{-1}(\{4\})=\{-2,2\}, \qquad f^{-1}(\{-1\})=\varnothing.

Thus an output may have several inputs producing it, or none. More generally,

y\in f(X)\quad\Longleftrightarrow\quad f^{-1}(\{y\})\neq\varnothing,

an immediate restatement of the image definition. (jirka.org)

Behavior under set operations

Images preserve inclusion and arbitrary unions. If $A\subseteq B\subseteq X$ , then

f(A)\subseteq f(B).

For any indexed family $(A_i)_{i\in I}$ of subsets of $X$ ,

f\!\left(\bigcup_{i\in I}A_i\right) =\bigcup_{i\in I}f(A_i).

In particular, the image of the empty set is empty. These identities follow because an input belongs to a union exactly when it belongs to at least one member of the family. (ncatlab.org)

Images do not generally preserve intersections:

f(A\cap B)\subseteq f(A)\cap f(B),

but equality can fail. For example, under $f(x)=x^2$ , the disjoint sets $A=\{-1\}$ and $B=\{1\}$ both have image $\{1\}$ , while their intersection has empty image. The failure occurs because the same output can come from different inputs. For an injective function, equality holds for every pair of subsets. (ncatlab.org)

Images also behave naturally under function composition. For $f:X\to Y$ , $g:Y\to Z$ , and $A\subseteq X$ ,

(g\circ f)(A)=g(f(A)).

Consequently, the image of a composite is contained in the image of its outer function. This identity follows by substituting $(g\circ f)(x)=g(f(x))$ into the definition of image. (math.dartmouth.edu)

Images in linear algebra

For a linear map $T:V\to W$ between vector spaces, the image of the linear map is a linear subspace of $W$ . When the map is represented by a matrix $M$ , its image is the span of the columns of $M$ :

\operatorname{im}(M)=\{Mx:x\in\mathbb R^n\}.

Its dimension is the rank of $M$ . (math.mit.edu)

The image determines which right-hand sides of a linear system are attainable:

Mx=b\text{ has a solution} \quad\Longleftrightarrow\quad b\in\operatorname{im}(M).

The rank–nullity theorem relates the dimension of this image to the dimension of the null space:

\dim\operatorname{im}(M)+\dim\ker(M)=n.

Thus the image describes attainable outputs, while the null space describes inputs sent to zero. (math.mit.edu)

Images under continuous functions

In topology, a continuous function preserves certain properties when taking images:

The continuous image of a compact space is compact.
The continuous image of a connected space is connected.

These statements concern images equipped with the subspace topology inherited from the codomain. (math.toronto.edu)

Because connected subsets of the real line are intervals, the image of an interval under a continuous real-valued function is again an interval. This is closely related to the intermediate value theorem. On a nonempty closed bounded interval, compactness additionally guarantees that the image has an attained minimum and maximum; hence it is a closed bounded interval, possibly consisting of one point. This connects the image concept with the extreme value theorem. (math.toronto.edu)

References

Basic Analysis Ijirka.org
Introduction (CS 2800, Fall 2016)cs.cornell.edu
Geneseo Math 239 01 Functions Introtildesites.geneseo.edu
AATA Sets and Equivalence Relationsmath.dartmouth.edu
Interactions of images and pre-images with unions and intersections in nLabncatlab.org
03 Lecture Notes, Spring 2025math.mit.edu
MAT246: Outline of Point Set Topologymath.toronto.edu