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A function f from X to Y. The blue oval Y is the codomain of f. The yellow oval inside Y is the image of f, and the red oval X is the domain of f. In mathematics, a codomain or set of destination of a function is a set into which all of the output of the function is constrained to fall. It is the set Y in the notation f: X → Y.
Given two sets X and Y, a binary relation f between X and Y is a function (from X to Y) if for every element x in X there is exactly one y in Y such that f relates x to y. The sets X and Y are called the domain and codomain of f, respectively. The image of the function f is the subset of Y consisting of only those elements y of Y such that ...
A function :, with domain X and codomain Y, is bijective, if for every y in Y, there is one and only one element x in X such that y = f(x). In this case, the inverse function of f is the function f − 1 : Y → X {\displaystyle f^{-1}:Y\to X} that maps y ∈ Y {\displaystyle y\in Y} to the element x ∈ X {\displaystyle x\in X} such that y = f ...
A function f from X to Y. The set of points in the red oval X is the domain of f. Graph of the real-valued square root function, f(x) = √ x, whose domain consists of all nonnegative real numbers. In mathematics, the domain of a function is the set of inputs accepted by the function.
The cardinality of the domain of a surjective function is greater than or equal to the cardinality of its codomain: If f : X → Y is a surjective function, then X has at least as many elements as Y, in the sense of cardinal numbers. (The proof appeals to the axiom of choice to show that a function g : Y → X satisfying f(g(y)) = y for all y ...
For every function f, let X be a subset of the domain and Y a subset of the codomain. One has always X ⊆ f −1 (f(X)) and f(f −1 (Y)) ⊆ Y, where f(X) is the image of X and f −1 (Y) is the preimage of Y under f. If f is injective, then X = f −1 (f(X)), and if f is surjective, then f(f −1 (Y)) = Y.
Even function: is symmetric with respect to the Y-axis. Formally, for each x: f (x) = f (−x). Odd function: is symmetric with respect to the origin. Formally, for each x: f (−x) = −f (x). Relative to a binary operation and an order: Subadditive function: for which the value of f (x + y) is less than or equal to f (x) + f (y).
The preimage of element ′ is . is a function from domain to codomain . The image of all elements in subset A {\displaystyle A} is subset B {\displaystyle B} . The preimage of B {\displaystyle B} is subset C {\displaystyle C} f {\displaystyle f} is a function from domain X {\displaystyle X} to codomain Y . {\displaystyle Y.}