Unformatted text preview: ve. Hence, a = −a, b = −b,
and ab = ab. The equation ab = ab 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 = ab reduces to −ab = a(−b) which is true. A symmetric argument shows the
equation ab = ab holds when a is negative and b is positive. Finally, if either a or b (or both)
are zero, then both sides of ab = ab are zero, and so the equation holds in this case, too. All
of this rhetoric has shown that the equation ab = ab 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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 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

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