# Image (mathematics)

In mathematics, the image of a function is the set of all output values it may take.

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 ". 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 : XY is a function from the set X to the set Y.

### Image of an element

If x is a member of X, then f(x) = y (the value of f when applied to x) is the image of x under f. y is alternatively known as the output of f for argument x.

### Image of a subset

The image of a subset AX under f is the subset f[A]Y defined by (using set-builder notation):

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

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[.] 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. See Notation below.

### Image of a function

The image of a function is the image of its entire domain.

### Generalization to binary relations

If R is an arbitrary binary relation on X×Y, the set { y∈Y | xRy for some xX } is called the image, or the range, of R. Dually, the set { xX | xRy for some y∈Y } is called the domain of R.

## Inverse image

Let f be a function from X to Y. The preimage or inverse image of a set BY under f is the subset of X defined by

${\displaystyle f^{-1}[B]=\{x\in X\,|\,f(x)\in B\}.}$

The inverse image of a singleton, denoted by f −1[{y}] or by f −1[y], is also called the fiber over y or the level set of y. The set of all the fibers over the elements of Y is a family of sets indexed by Y.

For example, for the function f(x) = x2, the inverse image of {4} would be {−2, 2}. Again, if there is no risk of confusion, denote f −1[B] by f −1(B), and think of f −1 as a function from the power set of Y to the power set of X. The notation f −1 should not be confused with that for inverse function. The notation coincides with the usual one, though, for bijections, in the sense that the inverse image of B under f is the image of B under f −1.

## Notation for image and inverse image

The traditional notations used in the previous section can be confusing. An alternative[1] is to give explicit names for the image and preimage as functions between powersets:

### Arrow notation

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

### Star notation

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

### Other terminology

• An alternative notation for f[A] used in mathematical logic and set theory is f "A.[2][3]
• Some texts refer to the image of f as the range of f, but this usage should be avoided because the word "range" is also commonly used to mean the codomain of f.

## Examples

1. f: {1, 2, 3} → {a, b, c, d} defined by ${\displaystyle f(x)=\left\{{\begin{matrix}a,&{\mbox{if }}x=1\\a,&{\mbox{if }}x=2\\c,&{\mbox{if }}x=3.\end{matrix}}\right.}$
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 {1, 2}. The preimage of {b, d} is the empty set {}.
2. f: RR defined by f(x) = x2.
The image of {−2, 3} under f is f({−2, 3}) = {4, 9}, and the image of f is R+. The preimage of {4, 9} under f is f −1({4, 9}) = {−3, −2, 2, 3}. The preimage of set N = {nR | n < 0} under f is the empty set, because the negative numbers do not have square roots in the set of reals.
3. f: R2R defined by f(x, y) = x2 + y2.
The fibres f −1({a}) are concentric circles about the origin, the origin itself, and the empty set, depending on whether a > 0, a = 0, or a < 0, respectively.
4. If M is a manifold and π: TMM is the canonical projection from the tangent bundle TM to M, then the fibres of π are the tangent spaces Tx(M) for xM. This is also an example of a fiber bundle.
5. A quotient group is a homomorphic image.

## Properties

Counter-examples based on
f:→ℝ, xx2, showing
that equality generally need
not hold for some laws:

For every function f : XY, all subsets A, A1, and A2 of X and all subsets B, B1, and B2 of Y, the following properties hold:

• f(A1  A2) = f(A1)  f(A2)[4]
• f(A1  A2) f(A1)  f(A2)[4]
• f(A  f −1(B)) = f(A)  B
• f −1(B1  B2) = f −1(B1)  f −1(B2)
• f −1(B1  B2) = f −1(B1)  f −1(B2)
• f(A) = ∅ ⇔ A = ∅
• f −1(B) = ∅ ⇔ B  (f(X))C
• f(A)  B = ∅ ⇔ A  f −1(B) = ∅
• f(A)  BA   f −1(B)
• B  f(A) ⇔ ${\displaystyle \exists }$C  A (f(C) = B)
• f(f −1(B))  B[5]
• f −1(f(A))  A[6]
• f(f −1(B)) = B  f(X)
• f −1(f(X)) = X
• A1A2f(A1) ⊆ f(A2)
• B1B2f −1(B1) ⊆ f −1(B2)
• f −1(BC) = (f −1(B))C
• (f |A)−1(B) = Af −1(B).

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:

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

(Here, S can be infinite, even uncountably infinite.)

With respect to the algebra of subsets, by the above the inverse image function is a lattice homomorphism while the image function is only a semilattice homomorphism (it does not always preserve intersections).

## Notes

1. Blyth 2005, p. 5
2. Jean E. Rubin (1967). Set Theory for the Mathematician. Holden-Day. p. xix. ASIN B0006BQH7S.
3. M. Randall Holmes: Inhomogeneity of the urelements in the usual models of NFU, December 29, 2005, on: Semantic Scholar, p. 2
4. Kelley (1985), p. 85
5. Equality holds if B is a subset of f(X) or, in particular, if f is surjective. See Munkres, J.. Topology (2000), p. 19.
6. Equality holds if f is injective. See Munkres, J.. Topology (2000), p. 19.

## References

• Artin, Michael (1991). Algebra. Prentice Hall. ISBN 81-203-0871-9
• T.S. Blyth, Lattices and Ordered Algebraic Structures, Springer, 2005, ISBN 1-85233-905-5.
• Munkres, James R. (2000). Topology (2 ed.). Prentice Hall. ISBN 978-0-13-181629-9.
• Kelley, John L. (1985). General Topology. Graduate Texts in Mathematics. 27 (2 ed.). Birkhäuser. ISBN 978-0-387-90125-1.