Functions:Forward Image

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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.

Definition

The word "image" is used in three related ways. In these definitions, is a function from the set to the set .

Image of an element

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 "take the value " or "take as a value" if there exists some in the function's domain such that . Similarly, given a set , is said to "take a value in " if there exists some in the function's domain such that . However, " takes all values in " and " is valued in " means that for every point in 's domain.

Image of a subset

The image of a subset under , denoted , is the subset of which can be defined using set-builder notation as follows:

When there is no risk of confusion, is simply written as . This convention is a common one; the intended meaning must be inferred from the context. This makes a function whose domain is the power set of (the set of all subsets of ), and whose codomain is the power set of .

Image of a function

The image of a function is the image of its entire domain, also known as the range of the function.[3] This usage should be avoided because the word "range" is also commonly used to mean the codomain of {\displaystyle f.}f.

Generalization to binary relations

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 .

Examples

1. defined by

The image of the set under is The image of the function is The preimage of is The preimage of is also The preimage of is the empty set

2. defined by

The image of under is and the image of is (the set of all positive real numbers and zero). The preimage of under is The preimage of set under is the empty set, because the negative numbers do not have square roots in the set of reals.

3. defined by

The fiber are concentric circles]] about the origin, the origin itself, and the empty set, depending on whether respectively. (if then the fiber is the set of all satisfying the equation of the origin-concentric ring )

4. If is a manifold and is the canonical projection from the tangent bundle to then the fibers of are the tangent spaces This is also an example of a fiber bundle.

5. A quotient group is a homomorphic image.

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