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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 some cases the codomain and the image of a function are the same set; such a function is called surjective or onto.
The range, T, has the cumulative distribution function [3] [4] = [(+) ()].Gumbel notes that the "beauty of this formula is completely marred by the facts that, in general, we cannot express G(x + t) by G(x), and that the numerical integration is lengthy and tiresome."
This function is unusual because it actually attains the limiting values of -1 and 1 within a finite range, meaning that its value is constant at -1 for all and at 1 for all . Nonetheless, it is smooth (infinitely differentiable, C ∞ {\displaystyle C^{\infty }} ) everywhere , including at x = ± 1 {\displaystyle x=\pm 1} .
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 .
The function f is surjective (or onto, or is a surjection) if its range () equals its codomain , that is, if, for each element of the codomain, there exists some element of the domain such that () = (in other words, the preimage () of every is nonempty).
Say (,) is equipped with its usual topology. Then the essential range of f is given by . = { >: < {: | | <}}. [7]: Definition 4.36 [8] [9]: cf. Exercise 6.11 In other words: The essential range of a complex-valued function is the set of all complex numbers z such that the inverse image of each ε-neighbourhood of z under f has positive measure.
The standard logistic function is the logistic function with parameters =, =, =, which yields = + = + = / / + /.In practice, due to the nature of the exponential function, it is often sufficient to compute the standard logistic function for over a small range of real numbers, such as a range contained in [−6, +6], as it quickly converges very close to its saturation values of 0 and 1.
Range of a function, a set containing the output values produced by a function; Range (statistics), the difference between the highest and the lowest values in a set; Interval (mathematics), also called range, a set of real numbers that includes all numbers between any two numbers in the set; Column space, also called the range of a matrix, is ...