# Image (mathematics)

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

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* ". Similarly, 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.

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*.

If *x* is a member of *X*, then the image of *x* under *f*, denoted *f*(*x*),^{[1]} is the value of *f* when applied to *x.* *f*(*x*) is alternatively known as the output of *f* for argument *x*.

The *image* of a function is the image of its entire domain, also known as the range of the function.^{[4]}

The traditional notations used in the previous section can be confusing. An alternative^{[7]} is to give explicit names for the image and preimage as functions between power sets:

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:

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

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