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The Absolute Value Function, and its Properties
The Absolute Value Function, and its PropertiesOne of the most used functions in mathematics is the Absolute Value function. Its definition andsome of its Properties are given Value FunctionThe Absolute Value of a real numberx,|x|, is|x|={xifx 0 xifx <0 The graph of the Absolute Value function is shown belowxyExample 1|2|= 2,| 2|= ( 2) = 2 The Absolute Value function is used to measure the distance between two numbers. Thus, thedistance betweenxand 0 is|x 0|=|x|, and the distance betweenxandyis|x y|. Thus, thedistance from 2 to 4 is| 2 ( 4)|=| 2 + 4|=|2|= 2, and the distance from 2 to 5 is| 2 5|=| 7|= following Properties of the Absolute Value function need to be any two real numbersxandy, we have|xy|=|x| |y|.This equality can be verified by considering cases. One of the four possible cases is checkedas follows: Supposex <0 andy 0. Thenxyis 0 and we have|xy|= (xy) = ( x)y=|x| |y|.The other three cases are similarly 2 For any real numberx, and any nonnegative numbera, we have|x| aif and only if a x is verified by considering the two possible casesx 0 andx <0.}
The Absolute Value Function, and its Properties One of the most used functions in mathematics is the absolute value function. Its de nition and some of its properties are given below. Absolute Value Function The absolute value of a real number x, jxj, is jxj=
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