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单词 SumAndProductAndQuotientOfFunctions
释义

sum and product and quotient of functions


Let A be a set and M a left R-module.  If  f:AM  and  g:AM,  then one may define the sum of functions f and g as the following functionMathworldPlanetmathf+g:AM:

(f+g)(x):=f(x)+g(x)xA

If r is any element of the ring R, then the scalar-multiplied functionrf:AM  is defined as

(rf)(x):=rf(x)xA.

Let A again be a set and K a field or a skew field.  If  f:AK  and  g:AK,  then one can define the product of functions f and g as the function  fg:AK as follows:

(fg)(x):=f(x)g(x)xA

The quotient of functions f and g is the function fg:{aAg(a)0}K  defined as

fg(x):=f(x)g(x)xA{aAg(a)=0}.

In particular, the incremental quotient of functions f(y)-f(x)y-x, as y tends to x, gave rise to the important concept of derivative. As another example, we can with a conscience say that the tangent (http://planetmath.org/TrigonometricFunction) function is the quotient of the sine (http://planetmath.org/TrigonometricFunction) and the cosine functions.

Titlesum and product and quotient of functions
Canonical nameSumAndProductAndQuotientOfFunctions
Date of creation2013-03-22 17:44:24
Last modified on2013-03-22 17:44:24
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id12
Authorpahio (2872)
Entry typeDefinition
Classificationmsc 03E20
Related topicDirectSumOfEvenoddFunctionsExample
Related topicLimitRulesOfFunctions
Related topicPolynomialFunction
Related topicProofOfLimitRuleOfProduct
Related topicContinuousDerivativeImpliesBoundedVariation
Related topicPropertiesOfRiemannStieltjesIntegral
Related topicInfimumAndSupremumOfSumAndProduct
Related topicPropertiesOfVectorValuedFunctio
Definessum of functions
Definesproduct of functions
Definesquotient of functions
Definesscalar-multiplied function
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更新时间:2025/5/4 13:02:33