Chapter 1 | Functions and Graphs
97
In Figure 1.43 , we graph both y = x 2 and y =2 x to show how the graphs differ.
Figure 1.43 Both 2 x and x 2 approach infinity as x →∞, but 2 x grows more rapidly than x 2 . As x →−∞, x 2 →∞, whereas 2 x →0.
Evaluating Exponential Functions Recall the properties of exponents: If x is a positive integer, then we define b x = b · b ⋯ b (with x factors of b ). If x is a negative integer, then x =− y for some positive integer y , and we define b x = b − y =1/ b y . Also, b 0 is defined to be 1. If x is a rational number, then x = p / q , where p and q are integers and b x = b p / q = b p q . For example, 9 3/2 = 9 3 =27. However, how is b x defined if x is an irrational number? For example, what do we mean by 2 2 ? This is too complex a question for us to answer fully right now; however, we can make an approximation. In Table 1.11 , we list some rational numbers approaching 2, and the values of 2 x for each rational number x are presented as well. We claim that if we choose rational numbers x getting closer and closer to 2, the values of 2 x get closer and closer to some number L . We define that number L to be 2 2 .
1.414213
1.4
1.41
1.414
1.4142
1.41421
x
2 x
2.639 2.65737 2.66475 2.665119 2.665138 2.665143
Table 1.11 Values of 2 x for a List of Rational Numbers Approximating 2
Example 1.33 Bacterial Growth
Suppose a particular population of bacteria is known to double in size every 4 hours. If a culture starts with 1000 bacteria, the number of bacteria after 4 hours is n (4) = 1000 · 2. The number of bacteria after 8 hours is n (8) = n (4)·2 =1000·2 2 . In general, the number of bacteria after 4 m hours is n (4 m ) =1000·2 m . Letting
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