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Given its domain and its codomain, a function is uniquely represented by the set of all pairs (x, f (x)), called the graph of the function, a popular means of illustrating the function. [note 1] [4] When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinates of a point in the plane.
Consider a quartic equation expressed in the form + + + + =: . There exists a general formula for finding the roots to quartic equations, provided the coefficient of the leading term is non-zero.
For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth (1/5 or 0.2), and the reciprocal of 0.25 is 1 divided by 0.25, or 4. The reciprocal function, the function f(x) that maps x to 1/x, is one of the simplest examples of a function which is its own inverse (an involution).
In mathematics, a free module is a module that has a basis, that is, a generating set that is linearly independent.Every vector space is a free module, [1] but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules.
In complex analysis, a branch of mathematics, the Koebe 1/4 theorem states the following: Koebe Quarter Theorem. The image of an injective analytic function f : D → C {\displaystyle f:\mathbf {D} \to \mathbb {C} } from the unit disk D {\displaystyle \mathbf {D} } onto a subset of the complex plane contains the disk whose center is f ( 0 ...
Given real numbers x and y, integers m and n and the set of integers, floor and ceiling may be defined by the equations ⌊ ⌋ = {}, ⌈ ⌉ = {}. Since there is exactly one integer in a half-open interval of length one, for any real number x, there are unique integers m and n satisfying the equation
In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “universal” group having these properties, in the sense that any two homomorphisms from G and H into a group K factor uniquely through a ...
A descent datum consists roughly of a covering of an object V of C by a family V i, elements x i in the fiber over V i, and morphisms f ji between the restrictions of x i and x j to V ij =V i × V V j satisfying the compatibility condition f ki = f kj f ji. The descent datum is called effective if the elements x i are essentially the pullbacks ...