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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.
The term domain is also commonly used in a different sense in mathematical analysis: a domain is a non-empty connected open set in a topological space. In particular, in real and complex analysis , a domain is a non-empty connected open subset of the real coordinate space R n {\displaystyle \mathbb {R} ^{n}} or the complex coordinate space C n ...
The image of a function is the image of its entire domain, also known as the range of the function. [3] This last usage should be avoided because the word "range" is also commonly used to mean the codomain of f . {\displaystyle f.}
Sigmoid functions, particularly the logistic function, have a domain of all real numbers and typically produce output values in the range from 0 to 1, although some variations, like the hyperbolic tangent, output values between −1 and 1.
A codomain is part of a function f if f is defined as a triple (X, Y, G) where X is called the domain of f, Y its codomain, and G its graph. [1] The set of all elements of the form f(x), where x ranges over the elements of the domain X, is called the image of f. The image of a function is a subset of its codomain so it might not coincide with it.
Graph of a linear function Graph of a polynomial function, here a quadratic function. Graph of two trigonometric functions: sine and cosine. A real function is a real-valued function of a real variable, that is, a function whose codomain is the field of real numbers and whose domain is a set of real numbers that contains an interval.
Interpretation for surjective functions in the Cartesian plane, defined by the mapping f : X → Y, where y = f(x), X = domain of function, Y = range of function. Every element in the range is mapped onto from an element in the domain, by the rule f. There may be a number of domain elements which map to the same range element.
More generally, the restriction (or domain restriction or left-restriction) of a binary relation between and may be defined as a relation having domain , codomain and graph ( ) = {(,) ():}. Similarly, one can define a right-restriction or range restriction R B . {\displaystyle R\triangleright B.}