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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.
It was originally known as "HECKE and Manin". After a short while it was renamed SAGE, which stands for ‘’Software of Algebra and Geometry Experimentation’’. Sage 0.1 was released in 2005 and almost a year later Sage 1.0 was released. It already consisted of Pari, GAP, Singular and Maxima with an interface that rivals that of Mathematica.
Size of this JPG preview of this PDF file: ... This work is in the public domain in the United States ... Calculus Made Easy.pdf/1; Page:Calculus Made Easy.pdf/3; ...
Let F be a field and let X be any set. The functions X → F can be given the structure of a vector space over F where the operations are defined pointwise, that is, for any f, g : X → F, any x in X, and any c in F, define (+) = + () = When the domain X has additional structure, one might consider instead the subset (or subspace) of all such functions which respect that structure.
For example, in calculus if is a differentiable function defined on some interval, then it is sufficient to show that the derivative is always positive or always negative on that interval. In linear algebra, if f {\displaystyle f} is a linear transformation it is sufficient to show that the kernel of f {\displaystyle f} contains only the zero ...
In calculus, and especially multivariable calculus, the mean of a function is loosely defined as the average value of the function over its domain. In one variable, the mean of a function f(x) over the interval (a,b) is defined by: [1] ¯ = ().
One can define the cokernel in the general framework of category theory.In order for the definition to make sense the category in question must have zero morphisms.The cokernel of a morphism f : X → Y is defined as the coequalizer of f and the zero morphism 0 XY : X → Y.