Range of a function

Subset of a function's codomain

In mathematics, the range of a function may refer to either of two closely related concepts: the codomain of the function, or the image of the function. In some cases the codomain and the image of a function are the same set; such a function is called surjective or onto. For any non-surjective function f : X → Y , {\displaystyle f:X\to Y,} the codomain Y {\displaystyle Y} and the image Y ~ {\displaystyle {\tilde {Y}}} are different; however, a new function can be defined with the original function's image as its codomain, f ~ : X → Y ~ {\displaystyle {\tilde {f}}:X\to {\tilde {Y}}} where f ~ ( x) = f ( x) .

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