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 at each element of a given subset of its domain produces a set, called the "image of under (or through) ". Similarly, the inverse image (or preimage) of a given subset of the codomain of is the set of all elements of the domain that map to the members of 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, is a function from the set to the set If is a member of then the image of under denoted is the value of when applied to is alternatively known as the output of for argument Given the function is said to "" or "" if there exists some in the function's domain such that Similarly, given a set is said to "" if there exists in the function's domain such that However, "" and "" means that for point in 's domain. Throughout, let be a function. The under of a subset of is the set of all for It is denoted by or by when there is no risk of confusion. Using set-builder notation, this definition can be written as This induces a function where denotes the power set of a set that is the set of all subsets of See below for more. The image of a function is the image of its entire domain, also known as the range of the function. This last usage should be avoided because the word "range" is also commonly used to mean the codomain of If is an arbitrary binary relation on then the set is called the image, or the range, of Dually, the set is called the domain of Let be a function from to The preimage or inverse image of a set under denoted by is the subset of defined by Other notations include and The inverse image of a singleton set, denoted by or by is also called the fiber or fiber over or the level set of The set of all the fibers over the elements of is a family of sets indexed by For example, for the function the inverse image of would be Again, if there is no risk of confusion, can be denoted by and can also be thought of as a function from the power set of to the power set of The notation should not be confused with that for inverse function, although it coincides with the usual one for bijections in that the inverse image of under is the image of under The traditional notations used in the previous section do not distinguish the original function from the image-of-sets function ; likewise they do not distinguish the inverse function (assuming one exists) from the inverse image function (which again relates the powersets).