# Image (mathematics)

In mathematics, the **image** of a set under a function is the set of all outputs the function produces when applied to the elements of that set. For a function \( f : A \to B \) and a subset \( C \subseteq A \), the image of \( C \) is \( f(C) = \{ f(x) : x \in C \} \), a subset of \( B \). The companion notion is the **preimage** (or inverse image): for \( D \subseteq B \), \( f^{-1}(D) = \{ x \in A : f(x) \in D \} \), the set of all inputs that land in \( D \)<sup>[1](https://personal.math.ubc.ca/~PLP/book/ch-functions-5.html)</sup>. The image of the whole domain, \( f(A) \), is called the image or range of the function<sup>[2](https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/An_Introduction_to_Proof_via_Inquiry-Based_Learning_%28Ernst%29/08%3A_New_Page/8.04%3A_New_Page)</sup>.

| Key fact | Detail |
|---|---|
| Definition | For \( f : A \to B \) and \( C \subseteq A \), the image is \( f(C) = \{ f(x) : x \in C \} \subseteq B \); for \( D \subseteq B \), the preimage is \( f^{-1}(D) = \{ x \in A : f(x) \in D \} \)<sup>[1](https://personal.math.ubc.ca/~PLP/book/ch-functions-5.html)</sup> |
| Image of the domain | \( f(A) \) is the range (image) of the function, always a subset of the codomain \( B \)<sup>[2](https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/An_Introduction_to_Proof_via_Inquiry-Based_Learning_%28Ernst%29/08%3A_New_Page/8.04%3A_New_Page)</sup> |
| Preimage needs no inverse | \( f^{-1}(D) \) is defined for every function; the symbol \( f^{-1} \) here is set-valued and does not require an inverse function to exist<sup>[3](https://ogbonlab.com/textbooks/introduction-to-mathematical-proof/chapter-5-functions/images-and-inverse-images)</sup> |
| Round-trip inclusions | \( A \subseteq f^{-1}(f(A)) \) and \( f(f^{-1}(C)) \subseteq C \) always hold; equality can fail<sup>[4](https://courses.maths.ox.ac.uk/mod/page/view.php?id=65663)</sup> |
| Set operations | Images preserve unions but not in general intersections; preimages preserve unions, intersections, and complements<sup>[5](https://ncatlab.org/nlab/show/interactions%2Bof%2Bimages%2Band%2Bpre-images%2Bwith%2Bunions%2Band%2Bintersections)</sup> |
| Fibers | The preimage of a singleton, \( f^{-1}(\{y\}) \), may be empty, a single point, or larger; injectivity means every fiber has at most one element, surjectivity means every fiber has at least one<sup>[6](https://math.berkeley.edu/~wodzicki/H104.F10/Cardinality.pdf)</sup> |
| Terminology | "Range" is used inconsistently in the literature, sometimes meaning the image and sometimes the codomain<sup>[7](https://www.cs.cornell.edu/courses/cs2800/2017fa/lectures/lec12-jectivity.html)</sup> |

## Definitions and notation

The element-level notion comes first: if \( f(a) = b \), then \( b \) is called the image of \( a \) under \( f \). Extending this to sets, the image of \( C \subseteq A \) collects all values \( f(x) \) with \( x \in C \), and the preimage of \( D \subseteq B \) collects all \( x \in A \) with \( f(x) \in D \)<sup>[1](https://personal.math.ubc.ca/~PLP/book/ch-functions-5.html)</sup>. Besides \( f(C) \), the image of the whole domain is also written \( \operatorname{im} f \)<sup>[8](https://sites.math.northwestern.edu/~scanez/courses/300/notes/functions.pdf)</sup>.

The preimage notation \( f^{-1} \) is a frequent source of confusion. It is defined regardless of whether \( f \) has an inverse function; the inverse relation \( f^{-1} \) is a function exactly when \( f \) is a bijection, but \( f^{-1}(D) \) as a set makes sense for any \( f \)<sup>[2](https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/An_Introduction_to_Proof_via_Inquiry-Based_Learning_%28Ernst%29/08%3A_New_Page/8.04%3A_New_Page)</sup>. Writing \( f^{-1}(y) \) for \( f^{-1}(\{y\}) \) is a common abuse of notation, since strictly the preimage is taken of a set; some texts have tried alternative notation such as \( f^{-|}(C) \) to avoid confusion with inverse functions, without wide adoption<sup>[9](https://api.pageplace.de/preview/DT0400.9780191568305_A35506072/preview-9780191568305_A35506072.pdf)</sup>.

## Worked example

Take \( f : \mathbb{R} \to \mathbb{R} \) with \( f(x) = x^2 \). Then \( f^{-1}(\{0\}) = \{0\} \), \( f^{-1}(\{1\}) = \{-1, 1\} \), and \( f^{-1}(\{-1\}) = \varnothing \), since no real number squares to \( -1 \)<sup>[1](https://personal.math.ubc.ca/~PLP/book/ch-functions-5.html)</sup>. This shows that the preimage of a single point can contain zero, one, two, or even infinitely many elements<sup>[1](https://personal.math.ubc.ca/~PLP/book/ch-functions-5.html)</sup>.

The same function shows why images and preimages do not round-trip exactly. For \( A = \{1\} \), \( f^{-1}(f(A)) = \{-1, 1\} \neq A \), because \( -1 \) also maps into \( f(A) \)<sup>[4](https://courses.maths.ox.ac.uk/mod/page/view.php?id=65663)</sup>. And for \( C = \{-1\} \), \( f^{-1}(C) = \varnothing \), so \( f(f^{-1}(C)) = \varnothing \neq C \)<sup>[4](https://courses.maths.ox.ac.uk/mod/page/view.php?id=65663)</sup>.

## Behavior under set operations

Both operations interact with unions, intersections, and complements, but not symmetrically. For an indexed family \( \{S_i\} \) of subsets of the domain<sup>[5](https://ncatlab.org/nlab/show/interactions%2Bof%2Bimages%2Band%2Bpre-images%2Bwith%2Bunions%2Band%2Bintersections)</sup>:

\[ f\left( \bigcup_{i \in I} S_i \right) = \bigcup_{i \in I} f(S_i), \qquad f\left( \bigcap_{i \in I} S_i \right) \subseteq \bigcap_{i \in I} f(S_i). \]

The inclusion for intersections is in general proper. The standard counterexample uses \( f(x) = x^2 \) with \( A = [0, \infty) \) and \( B = (-\infty, 0) \): then \( A \cap B = \varnothing \) so \( f(A \cap B) = \varnothing \), while \( f(A) \cap f(B) = (0, \infty) \)<sup>[10](https://www.math.stonybrook.edu/Courses/MAT511/202606/MAT511Sum-Note.pdf)</sup>. The failure occurs because two different elements, one in each set, can map to the same output<sup>[11](https://sites.math.northwestern.edu/~scanez/courses/300/notes/lecture-notes-300.pdf)</sup>.

**Preimages are better behaved.** For \( \{T_i\} \subseteq B \)<sup>[5](https://ncatlab.org/nlab/show/interactions%2Bof%2Bimages%2Band%2Bpre-images%2Bwith%2Bunions%2Band%2Bintersections)</sup>:

\[ f^{-1}\left( \bigcup_{i \in I} T_i \right) = \bigcup_{i \in I} f^{-1}(T_i), \qquad f^{-1}\left( \bigcap_{i \in I} T_i \right) = \bigcap_{i \in I} f^{-1}(T_i). \]

Preimages also preserve complements: \( f^{-1}(X - Y) = f^{-1}(X) - f^{-1}(Y) \) for subsets \( X, Y \) of the codomain<sup>[11](https://sites.math.northwestern.edu/~scanez/courses/300/notes/lecture-notes-300.pdf)</sup>. In fact the preimage map \( f^* : \mathcal{P}(B) \to \mathcal{P}(A) \) preserves the whole [Boolean algebra](https://www.edgepedia.org/boolean-algebra) structure of unions, intersections, and complements<sup>[12](https://mathisart.org/set_theory02)</sup>.

## Fibers, injectivity, and surjectivity

For \( y \in B \), the set \( f^{-1}(\{y\}) = \{ x \in X : f(x) = y \} \) is called the fiber of \( f \) at \( y \)<sup>[6](https://math.berkeley.edu/~wodzicki/H104.F10/Cardinality.pdf)</sup>. Fibers restate the two basic properties of functions<sup>[6](https://math.berkeley.edu/~wodzicki/H104.F10/Cardinality.pdf)</sup>:

- \( f \) is injective if and only if every fiber has no more than one element.
- \( f \) is surjective if and only if every fiber has at least one element.

These conditions also control when the round-trip inclusions become equalities. If \( f \) is injective, then \( X = f^{-1}(f[X]) \) for every \( X \subseteq A \); if \( f \) is surjective, then \( f[f^{-1}[Y]] = Y \) for every \( Y \subseteq B \)<sup>[13](https://web.stanford.edu/class/archive/cs/cs103/cs103.1242/lectures/08/Condensed%20Slides.pdf)</sup>. A function is surjective precisely when its image equals its codomain<sup>[14](https://sites.millersville.edu/bikenaga/linear-algebra/functions/functions.html)</sup>.

## Terminology and history

The word "range" is a persistent ambiguity: some authors use it for the image, the set of values actually produced, while others use it for the codomain, the set into which outputs are constrained to fall<sup>[7](https://www.cs.cornell.edu/courses/cs2800/2017fa/lectures/lec12-jectivity.html)</sup>. For this reason some texts avoid "range" altogether<sup>[14](https://sites.millersville.edu/bikenaga/linear-algebra/functions/functions.html)</sup>.

Historically, the terminology goes back to [Richard Dedekind](https://www.edgepedia.org/richard-dedekind), who, writing in 1888, introduced a set of terms for the modern conception of a mapping that includes "image", identity map, composition of maps, and injective maps<sup>[15](https://mathshistory.st-andrews.ac.uk/Miller/mathsym/functions/)</sup>.

## Use in higher mathematics

Inverse images are a working tool well beyond set theory. In topology, where the idea of the inverse image of a set under a map is heavily used, continuity is defined by requiring the preimage of every open set to be open; the definition does not require an inverse function to exist<sup>[9](https://api.pageplace.de/preview/DT0400.9780191568305_A35506072/preview-9780191568305_A35506072.pdf)</sup>. The same pattern appears in measure theory, where a measurable mapping \( f : (X, S) \to (Y, J) \) between spaces equipped with \( \sigma \)-algebras or topologies is a map with \( T \in J \Rightarrow f^{-1}(T) \in S \)<sup>[16](https://www.sciencedirect.com/topics/mathematics/measurable-mapping)</sup>. In linear algebra, the image of a linear transformation \( T : V \to W \) is a subspace of \( W \), paired with the kernel as the two fundamental subspaces associated with \( T \)<sup>[17](https://www.math.ucla.edu/~tsmits/Teaching/115A/Handouts/lintransform.pdf)</sup>.

## References

1. [Section 10.3: Images and preimages of sets, University of British Columbia](https://personal.math.ubc.ca/~PLP/book/ch-functions-5.html)
2. [8.4: Images and Preimages of Functions, Mathematics LibreTexts](https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/An_Introduction_to_Proof_via_Inquiry-Based_Learning_%28Ernst%29/08%3A_New_Page/8.04%3A_New_Page)
3. [Images and Inverse Images, Interactive Mathematical Proof](https://ogbonlab.com/textbooks/introduction-to-mathematical-proof/chapter-5-functions/images-and-inverse-images)
4. [Introduction to University Mathematics: 3 Functions, Mathematical Institute, University of Oxford](https://courses.maths.ox.ac.uk/mod/page/view.php?id=65663)
5. [Interactions of images and pre-images with unions and intersections, nLab](https://ncatlab.org/nlab/show/interactions%2Bof%2Bimages%2Band%2Bpre-images%2Bwith%2Bunions%2Band%2Bintersections)
6. [Notes on Sets, Mappings, and Cardinality, A. Wodzicki, UC Berkeley](https://math.berkeley.edu/~wodzicki/H104.F10/Cardinality.pdf)
7. [Functions (injectivity and surjectivity), CS 2800, Cornell University](https://www.cs.cornell.edu/courses/cs2800/2017fa/lectures/lec12-jectivity.html)
8. [Notes on Functions, Northwestern University Math 300](https://sites.math.northwestern.edu/~scanez/courses/300/notes/functions.pdf)
9. [Direct and inverse images (textbook preview, Oxford University Press)](https://api.pageplace.de/preview/DT0400.9780191568305_A35506072/preview-9780191568305_A35506072.pdf)
10. [MAT 511: Fundamental Concepts of Math, Stony Brook University](https://www.math.stonybrook.edu/Courses/MAT511/202606/MAT511Sum-Note.pdf)
11. [Math 300 Lecture Notes, Northwestern University](https://sites.math.northwestern.edu/~scanez/courses/300/notes/lecture-notes-300.pdf)
12. [Images and preimages, mathIsART](https://mathisart.org/set_theory02)
13. [Set Theory Revisited, CS 103 lecture slides, Stanford University](https://web.stanford.edu/class/archive/cs/cs103/cs103.1242/lectures/08/Condensed%20Slides.pdf)
14. [Functions, linear algebra notes, Millersville University](https://sites.millersville.edu/bikenaga/linear-algebra/functions/functions.html)
15. [Earliest Uses of Function Symbols, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Miller/mathsym/functions/)
16. [Measurable Mapping, ScienceDirect Topics](https://www.sciencedirect.com/topics/mathematics/measurable-mapping)
17. [Linear Transformations, UCLA](https://www.math.ucla.edu/~tsmits/Teaching/115A/Handouts/lintransform.pdf)

---
*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Logic and discrete mathematics › Formal logic and foundations › Set theory › Elementary set theory*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license