Inverse image

In mathematics, an image is the subset of a function's codomain which is the output of the function from a subset of its domain.

Evaluating a function at each element of a subset X of the domain, produces a set called the image of X under or through the function. The inverse image or preimage of a particular subset S of the codomain of a function is the set of all elements of the domain that map to the members of S.

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 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.

The image of a subset A ⊆ X under f is the subset f[A] ⊆ Y defined by (in 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 the image of 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.

The image f[X] of the entire domain X of f is called simply the image of f.

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

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