# Image (mathematics)

# 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* : *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 (using 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 *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 *x*∈*X* } is called the image, or the range, of *R*. Dually, the set { *x*∈*X* | *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 *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, 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

with

with

Star notation

instead of

instead of

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

*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 {}.*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.*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.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.A quotient group is a homomorphic image.

Properties

Counter-examples based onf:ℝ→ℝ, x↦x^{2}, showingthat equality generally need not hold for some laws: |
---|

For every function *f* : *X* → *Y*, all subsets *A*, *A*1, and *A*2 of *X* and all subsets *B*, *B*1, and *B*2 of *Y*, the following properties hold:

*f*(*A*1 ∪*A*2) =*f*(*A*1) ∪*f*(*A*2)^{[4]}*f*(*A*1 ∩*A*2) ⊆*f*(*A*1) ∩*f*(*A*2)^{[4]}*f*(*A*∩*f*−1(*B*)) =*f*(*A*) ∩*B**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*) = ∅ ⇔*A*= ∅*f*−1(*B*) = ∅ ⇔*B*⊆ (*f*(*X*))C*f*(*A*) ∩*B*= ∅ ⇔*A*∩*f*−1(*B*) = ∅*f*(*A*) ⊆*B*⇔*A*⊆*f*−1(*B*)*B*⊆*f*(*A*) ⇔*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**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 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

## References

*Set Theory for the Mathematician*. Holden-Day. p. xix. ASIN B0006BQH7S.

*B*is a subset of

*f*(

*X*) or, in particular, if

*f*is surjective. See Munkres, J.. Topology (2000), p. 19.

*f*is injective. See Munkres, J.. Topology (2000), p. 19.