Stitz-Zeager_College_Algebra_e-book

1 f x x x 2 g x x 2 x 3 1 solution 1

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Unformatted text preview: finition, we have √ |5| = (5)2 = 25 = 5 and | − 5| = (−5)2 = 25 = 5. The long and short of both of these procedures is that |x| takes negative real numbers and assigns them to their positive counterparts while it leaves positive numbers alone. This last description is the one we shall adopt, and is summarized in the following definition. Definition 2.4. The absolute value of a real number x, denoted |x|, is given by |x| = −x, if x < 0 x, if x ≥ 0 In Definition 2.4, we define |x| using a piecewise-defined function. (See page 49 in Section 1.5.) To check that this definition agrees with what we previously understood as absolute value, note that since 5 ≥ 0, to find |5| we use the rule |x| = x, so |5| = 5. Similarly, since −5 < 0, we use the rule |x| = −x, so that | − 5| = −(−5) = 5. This is one of the times when it’s best to interpret the expression ‘−x’ as ‘the opposite of x’ as opposed to ‘negative x.’ Before we embark on studying absolute value functions, we remind ourselves of the properties of absolute value. Theo...
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