448 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions In this section, we extend the definition of ar to include all real (not just rational) values of the exponent r. Consider the graphs of y = 2x for different domains in Figure 13. y 2 –2 y = 2x; integers as domain 2 4 6 8 x y 2 –2 y = 2x; selected rational numbers as domain 8 6 4 2 x y –2 2 y = 2x; real numbers as domain 8 6 4 2 x 2!3 !3 Figure 13 The equations that use just integers or selected rational numbers as domain in Figure 13 leave holes in the graphs. In order for the graph to be continuous, we must extend the domain to include irrational numbers such as 23. We might evaluate 2 23 by approximating the exponent with the rational numbers 1.7, 1.73, 1.732, and so on. Because these values approach the value of 23 more and more closely, it is reasonable that 2 23 should be approximated more and more closely by the numbers 21.7, 21.73, 21.732, and so on. These expressions can be evaluated using rational exponents as follows. 21.7 = 217/10 = Q210 2 R 17 ≈3.249009585 Because any irrational number may be approximated more and more closely using rational numbers, we can extend the definition of ar to include all real number exponents and apply all previous theorems for exponents. In addition to the rules for exponents presented earlier, we use several new properties in this chapter. Additional Properties of Exponents For any real number a 70, a≠1, the following statements hold true. Property Description (a) ax is a unique real number for all real numbers x. y = a x can be considered a function ƒ1x2 = a x with domain 1-∞, ∞2. (b) ab =ac if and only if b =c. The function ƒ1x2 = ax is one-to-one. (c) If a +1 and m*n, then am *an. Example: 23 624 1a 712 Increasing the exponent leads to a greater number. The function ƒ1x2 = 2x is an increasing function. (d) If 0 *a *1 and m*n, then am +an. Example: A1 2B 2 7 A1 2B 3 10 6a 612 Increasing the exponent leads to a lesser number. The function ƒ1x2 = A1 2B x is a decreasing function.
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