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A function f from a set X to a set Y is an assignment of one element of Y to each element of X. The set X is called the domain of the function and the set Y is called the codomain of the function. If the element y in Y is assigned to x in X by the function f, one says that f maps x to y, and this is commonly written = ().
Given a function: from a set X (the domain) to a set Y (the codomain), the graph of the function is the set [4] = {(, ()):}, which is a subset of the Cartesian product.In the definition of a function in terms of set theory, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph.
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.
Composite function: is formed by the composition of two functions f and g, by mapping x to f (g(x)). Inverse function: is declared by "doing the reverse" of a given function (e.g. arcsine is the inverse of sine). Implicit function: defined implicitly by a relation between the argument(s) and the value.
A function : is injective if and only if is empty or is left-invertible; that is, there is a function : such that = identity function on X. Here, f ( X ) {\displaystyle f(X)} is the image of f {\displaystyle f} .
Note: If f takes its values in a ring (in particular for real or complex-valued f ), there is a risk of confusion, as f n could also stand for the n-fold product of f, e.g. f 2 (x) = f(x) · f(x). [11] For trigonometric functions, usually the latter is meant, at least for positive exponents. [11]
A constant function such as f(x) = π is a rational function since constants are polynomials. The function itself is rational, even though the value of f(x) is irrational for all x. Every polynomial function = is a rational function with () =
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.