a function (or mapping) from a set to a set is a rule that assigns to each element of exactly one element of . the set is called the domain of , and is called the range of . if assigns to , then is called the image of under . the subset of comprising all the images of elements of is called the image of under .
alternative definitions of functions
let be sets, and let be a property pertaining to an object and an object , such that for every , there is exactly one for which is true (this is sometimes known as the vertical line test). then we define the function defined by on the domain and codomain to be the object which, given any input , assigns an output , defined to be the unique object for which is true. thus, for any and ,
a function is a rule of assignment , together with a set that contains the image set of . the domain of the rule is also called the domain of the function ; the image set of is also called the image set of ; and the set is called the range of .