# Image (mathematics)

In mathematics, an **image** is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely evaluating the function at each element of a subset X of the domain produces a set called the image of X *under or through* the function. The **inverse image** or **preimage** of a particular subset *S* of the codomain of a function is the set of all elements of the domain that map to the members of *S*.

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* : *X* → *Y* 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 *A* ⊆ *X* under *f* is the subset *f*[*A*] ⊆ *Y* defined by (in set-builder notation):

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 the image of *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 *f*[*X*] of the entire domain *X* of *f* is called simply the image of *f*.

## Inverse image

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

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*) = *x*^{2}, the inverse image of {4} would be {-2,2}. Again, if there is no risk of confusion, we may 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 two coincide only if *f* is a bijection.

## 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

### Star notation

### Other terminology

- An alternative notation for
*f*[*A*] used in mathematical logic and set theory is*f*"*A*.^{[2]} - 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

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*: **R** → **R** defined by *f*(*x*) = *x*^{2}.

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*= {

*n*∈

**R**|

*n*< 0} under

*f*is the empty set, because the negative numbers do not have square roots in the set of reals.

3. *f*: **R**^{2} → **R** defined by *f*(*x*, *y*) = *x*^{2} + *y*^{2}.

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* :*TM*→*M* is the canonical projection from the tangent bundle *TM* to *M*, then the *fibres* of *π* are the tangent spaces *T*_{x}(*M*) for *x*∈*M*. This is also an example of a fiber bundle.

## Consequences

Given a function *f* : *X* → *Y*, for all subsets *A*, *A*_{1}, and *A*_{2} of *X* and all subsets *B*, *B*_{1}, and *B*_{2} of *Y* we have:

*f*(*A*_{1}∪*A*_{2}) =*f*(*A*_{1}) ∪*f*(*A*_{2})^{[3]}*f*(*A*_{1}∩*A*_{2}) ⊆*f*(*A*_{1}) ∩*f*(*A*_{2})^{[3]}*f*^{ −1}(*B*_{1}∪*B*_{2}) =*f*^{ −1}(*B*_{1}) ∪*f*^{ −1}(*B*_{2})*f*^{ −1}(*B*_{1}∩*B*_{2}) =*f*^{ −1}(*B*_{1}) ∩*f*^{ −1}(*B*_{2})*f*(A) ⊆*B*⇔*A*⊆*f*^{ −1}(*B*)*f*(*f*^{ −1}(*B*)) ⊆*B*^{[4]}*f*^{ −1}(*f*(*A*)) ⊇*A*^{[5]}*A*_{1}⊆*A*_{2}⇒*f*(*A*_{1}) ⊆*f*(*A*_{2})*B*_{1}⊆*B*_{2}⇒*f*^{ −1}(*B*_{1}) ⊆*f*^{ −1}(*B*_{2})*f*^{ −1}(*B*^{C}) = (*f*^{ −1}(*B*))^{C}- (
*f*|_{A})^{−1}(*B*) =*A*∩*f*^{ −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:

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

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

## See also

- Range (mathematics)
- Bijection, injection and surjection
- Kernel of a function
- Image (category theory)
- Set inversion

## Notes

- ↑ Blyth 2005, p. 5
- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
- ↑
^{3.0}^{3.1}Kelley (1985), Template:Google books quote - ↑ Equality holds if
*B*is a subset of Im(*f*) or, in particular, if*f*is surjective. See Munkres, J.. Topology (2000), p. 19. - ↑ Equality holds if
*f*is injective. See Munkres, J.. Topology (2000), p. 19.

## References

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- T.S. Blyth,
*Lattices and Ordered Algebraic Structures*, Springer, 2005, ISBN 1-85233-905-5. - {{#invoke:citation/CS1|citation

|CitationClass=book }}

- {{#invoke:citation/CS1|citation

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*This article incorporates material from Fibre on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*