Transcription of The Absolute Value Function, and its Properties
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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|.}
The following properties of the absolute value function need to be memorized. Lemma 1.For any two real numbers x and y, we have jxyj= jxjjyj. This equality can be veri ed by considering cases. One of the four possible cases is checked as follows: Suppose x < 0 and y 0. Then xy is 0 and we have jxyj= (xy) = ( x)y = jxjjyj.
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