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Function spaces play a fundamental role in advanced mathematical analysis, by allowing the use of their algebraic and topological properties for studying properties of functions. For example, all theorems of existence and uniqueness of solutions of ordinary or partial differential equations result of the study of function spaces.
At the same time, the mapping of a function to the value of the function at a point is a functional; here, is a parameter. Provided that f {\displaystyle f} is a linear function from a vector space to the underlying scalar field, the above linear maps are dual to each other, and in functional analysis both are called linear functionals .
In mathematics, some functions or groups of functions are important enough to deserve their own names. This is a listing of articles which explain some of these functions in more detail. There is a large theory of special functions which developed out of statistics and mathematical physics.
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.
is a function from domain X to codomain Y. The yellow oval inside Y is the image of . Sometimes "range" refers to the image and sometimes to the 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 mathematics, an argument of a function is a value provided to obtain the function's result. It is also called an independent variable. [1]For example, the binary function (,) = + has two arguments, and , in an ordered pair (,).
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.
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, and their inverses (e.g., arcsin, log, or x 1/n).