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In mathematics, for a function :, the image of an input value is the single output value produced by when passed . The preimage of an output value y {\displaystyle y} is the set of input values that produce y {\displaystyle y} .
For a continuous (see below for a definition) real-valued function which has a connected domain, the image is either an interval or a single value. In the latter case, the function is a constant function. The preimage of a given real number c is called a level set. It is the set of the solutions of the equation f(x 1, x 2, …, x n) = c.
In a category with all finite limits and colimits, the image is defined as the equalizer (,) of the so-called cokernel pair (,,), which is the cocartesian of a morphism with itself over its domain, which will result in a pair of morphisms ,:, on which the equalizer is taken, i.e. the first of the following diagrams is cocartesian, and the second equalizing.
This function maps each image to its unique preimage. The composition of two bijections is again a bijection, but if g ∘ f {\displaystyle g\circ f} is a bijection, then it can only be concluded that f {\displaystyle f} is injective and g {\displaystyle g} is surjective (see the figure at right and the remarks above regarding injections and ...
In mathematics, a surjective function (also known as surjection, or onto function / ˈ ɒ n. t uː /) is a function f such that, for every element y of the function's codomain, there exists at least one element x in the function's domain such that f(x) = y. In other words, for a function f : X → Y, the codomain Y is the image of the function ...
The preimage of y can be thought of as the image of y under the (multivalued) full inverse of the function f. The notion can be generalized to subsets of the range. Specifically, if S is any subset of Y , the preimage of S , denoted by f − 1 ( S ) {\displaystyle f^{-1}(S)} , is the set of all elements of X that map to S :
Let : be any function. If is any set then its preimage := under is necessarily an -saturated set.In particular, every fiber of a map is an -saturated set.. The empty set = and the domain = are always saturated.
In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable.