Section 2: Absolute Value

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 4.2 Absolute Value 353 Version: Fall 2007 4.2 Absolute Value Now that we have the fundamentals of piecewise-defined functions in place, we are ready to introduce the absolute value function. First, let’s state a trivial reminder of what it means to take the absolute value of a real number. In a sense, the absolute value of a number is a measure of its magnitude, sans (without) its sign. Thus, | 7 | = 7 and | − 7] = 7 . (1) Here is the formal definition of the absolute value of a real number. Definition 2. To find the absolute value of any real number, first locate the number on the real line. x 0 | x | The absolute value of the number is defined as its distance from the origin. For example, to find the absolute value of 7, locate 7 on the real line and then find its distance from the origin. 7 0 | 7 | = 7 To find the absolute value of 7 , locate 7 on the real line and then find its distance from the origin. 7 0 | − 7 | = 7 Some like to say that taking the absolute value “produces a number that is always positive.” However, this ignores an important exception, that is, | 0 | = 0 . (3) Copyrighted material. See: 1
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354 Chapter 4 Absolute Value Functions Version: Fall 2007 Thus, the correct statement is “the absolute value of any real number is either positive or it is zero,” i.e., the absolute value of a real number is “not negative.” 2 Instead of using the phrase “not negative,” mathematicians prefer the word “nonnegative.” When we take the absolute value of a number, the result is always nonnegative ; that is, the result is either positive or zero. In symbols, | x | ≥ 0 for all real numbers x . This makes perfect sense in light of Definition 2 . Distance is always nonnegative. However, the discussion above is not of sufficient depth to handle more sophisticated problems involving absolute value. A Piecewise Definition of Absolute Value Because absolute value is intimately connected with distance, mathematicians and sci- entists find it an invaluable tool for measurement and error analysis. However, we will need a formulaic definition of the absolute value if we want to use this tool in a mean- ingful way. We need to develop a piecewise definition of the absolute value function, one that will define the absolute value for any arbitrary real number x . We begin with a few observations. Remember, the absolute value of a number is always nonnegative (positive or zero). 1. If a number is negative, negating that number will make it positive. | − 5 | = ( 5) = 5 , and similarly, | − 12 | = ( 12) = 12 . Thus, if x < 0 (if x is negative), then | x | = x . 2. If x = 0 , then | x | = 0 . 3. If a number is positive, taking the absolute value of that number will not change a thing. | 5 | = 5 , and similarly, | 12 | = 12 . Thus, if x > 0 (if x is positive), then | x | = x . We can summarize these three cases with a piecewise definition .
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