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Chapter 1 | Functions and Graphs
t =4 m , we see that the number of bacteria after t hours is n ( t ) =1000·2 t /4 . Find the number of bacteria after 6 hours, 10 hours, and 24 hours. Solution The number of bacteria after 6 hours is given by n (6) = 1000 · 2 6/4 ≈2828 bacteria. The number of bacteria after 10 hours is given by n (10) = 1000 · 2 10/4 ≈5657 bacteria. The number of bacteria after 24 hours is given by n (24) = 1000 · 2 6 =64,000 bacteria.
Given the exponential function f ( x ) =100·3 x /2 , evaluate f (4) and f (10).
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Go to World Population Balance (http://www.openstax.org/l/20_exponengrow) for another example of exponential population growth.
Graphing Exponential Functions For any base b >0, b ≠1, the exponential function f ( x ) = b x is defined for all real numbers x and b x >0. Therefore, the domain of f ( x ) = b x is (−∞, ∞) and the range is (0, ∞). To graph b x , we note that for b >1, b x is increasing on (−∞, ∞) and b x →∞ as x →∞, whereas b x →0 as x →−∞. On the other hand, if 0< b <1, f ( x ) = b x is decreasing on (−∞, ∞) and b x →0 as x →∞ whereas b x →∞ as x →−∞ ( Figure 1.44 ).
Figure 1.44 If b >1, then b x is increasing on (−∞, ∞). If 0< b <1, then b x is decreasing on (−∞, ∞).
Visit this site (http://www.openstax.org/l/20_inverse) for more exploration of the graphs of exponential functions.
Note that exponential functions satisfy the general laws of exponents. To remind you of these laws, we state them as rules.
Rule: Laws of Exponents For any constants a >0, b >0, and for all x and y , 1. b x · b y = b x + y 2. b x b y = b x − y
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