# function (mathematics)

In mathematics, a function is a relation between two sets that assigns a unique value in one set (the output set) to every value in another set (the input set). The input set is called the domain of the function while the output set is called the codomain.

A function *f* that has domain *A* and codomain *B* is denoted:

This can be read as “*f* maps *A* into *B*” or “*f* is a function from *A* into *B*.”

If *a* is an element of the domain *A*, ie., , then is the unique element in the codomain *B* that *a* maps to. This value of *f* at *a* is called the image of *a* under *f*. The set of all image values is called the image of *f* or the range of *f*. The image of function *f* with domain *A* is denoted .

Note the difference between the codomain of *f* and the image of *f*. The codomain of a function is the set of all possible values that any member of the domain may map to. The image of a function is the set of actual values that the members of the domain *do* map to. For example a common codomain in mathematics is the set of real numbers , but a function may map to just a subset of the real numbers. In which case, the image of the function is a subset of the codomain .