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Note that if x 2 then x 3 so we replace x 3 with x

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Unformatted text preview: ve. Hence, |a| = −a, |b| = −b, and |ab| = ab. The equation |ab| = |a||b| becomes ab = (−a)(−b), which is true. Suppose a is positive and b is negative. Then ab is negative, and we have |ab| = −ab, |a| = a and |b| = −b. The equation |ab| = |a||b| reduces to −ab = a(−b) which is true. A symmetric argument shows the equation |ab| = |a||b| holds when a is negative and b is positive. Finally, if either a or b (or both) are zero, then both sides of |ab| = |a||b| are zero, and so the equation holds in this case, too. All of this rhetoric has shown that the equation |ab| = |a||b| holds true in all cases. The proof of the Quotient Rule is very similar, with the exception that b = 0. The Power Rule can be shown by repeated application of the Product Rule. The last three properties can be proved using Definition 2.4 and by looking at the cases when x ≥ 0, in which case |x| = x, or when x < 0, in which case |x| = −x. For example, if c > 0, and |x| = c, then if x ≥ 0, we have x = |x| = c. If, on the other hand, x < 0, then −x = |x| = c, so x = −c. The remaining properties are proved similarly and are left for the exercises. To graph functi...
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