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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.
In a strict sense, the composition g ∘ f is only meaningful if the codomain of f equals the domain of g; in a wider sense, it is sufficient that the former be an improper subset of the latter. [nb 1] Moreover, it is often convenient to tacitly restrict the domain of f, such that f produces only values in the domain of g.
An extension of a function f is a function g such that f is a restriction of g. A typical use of this concept is the process of analytic continuation, that allows extending functions whose domain is a small part of the complex plane to functions whose domain is almost the whole complex plane.
The convolution of f and g exists if f and g are both Lebesgue integrable functions in L 1 (R d), and in this case f∗g is also integrable (Stein & Weiss 1971, Theorem 1.3). This is a consequence of Tonelli's theorem. This is also true for functions in L 1, under the discrete convolution, or more generally for the convolution on any group.
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, counter-domain, 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.
with domain, the range of , sometimes denoted or (), [4] may refer to the codomain or target set (i.e., the set into which all of the output of is constrained to fall), or to (), the image of the domain of under (i.e., the subset of consisting of all actual outputs of ). The image of a function is always a subset of the codomain of the ...
For example, the square of the Fourier transform, W 2, is an intertwiner associated with J 2 = −I, and so we have (W 2 f)(x) = f (−x) is the reflection of the original function f. Complex domain [ edit ]
The function g : Y → X is said to be a right inverse of the function f : X → Y if f(g(y)) = y for every y in Y (g can be undone by f). In other words, g is a right inverse of f if the composition f o g of g and f in that order is the identity function on the domain Y of g.