1.4 Exponential Functions The funct Th fun c ti n exb^2x is called an e ponentia functi n b cause the va ia le, x, is the ex onent. It ho ld not b con use with the powe fun c tio t xb^ x2 in which th v ria le is t e base.I g neral, an e ponentia f n tion is f nct on o f the f In Appendix G we present an alterna- ormfe b bx hi mea h s re b is positive const nt. Le ’s r call what t tive approach to the exponential and If x^n, positiv integer, then logarithmic functions using integral calculus. bn n fact rs If x ^ 0, hen b0 1, an if x n , h re n is positiv int eger t 46 CHAPTER 1 Functions and Models If x is a rational number, x = p/q, where p and q are integers and q > 0, then bx = bp/q = sq bp = (sq b )p But what is the meaning of bx if x is an irrational number? For instance, what is meant by 2s3 or 57? To help us answer this question we first look at the graph of the function y = 2x, where x is rational. A representation of this graph is shown in Figure 1. We want to enlarge the domain of y = 2x to include both rational and irrational numbers. There are holes in the graph in Figure 1 corresponding to irrational values of x. We want to fill in the holes by defining f(x) = 2x, where x E R, so that f is an increasing function. In particular, since the irrational number s3 satisfies FIGURE 1 Representation of y = 2x, x rational 1.7 < s3 < 1.8 we must have 21.7 < 2s3 < 21.8 and we know what 21.7 and 21.8 mean because 1.7 and 1.8 are rational numbers. Similarly, if we use better approximations for s3 , we obtain better approximations for 2s3: 1.73 < s3 < 1.74 * 21.73 < 2s3 < 21.74 1.732 < s3 < 1.733 * 21.732 < 2s3 < 21.733 1.7320 < s3 < 1.7321 * 21.7320 < 2s3 < 21.7321 1.73205 < s3 < 1.73206 * 21.73205 < 2s3 < 21.73206 . . . . . . . . A proof of this fact is given in J. Marsden and A. Weinstein, Calculus Unlimited (Menlo Park, CA: Benjamin/Cummings, 1981). . . . . It can be shown that there is exactly one number that is greater than all of the numbers 21.7, 21.73, 21.732, 21.7320, 21.73205, . . . and less than all of the numbers 21.8, 21.74, 21.733, 21.7321, 21.73206, . . . We define 2s3 to be this number. Using the preceding approximation process we can compute it correct to six decimal places: 2s3 - 3.321997 Similarly, we can define 2x (or bx, if b > 0) where x is any irrational number. Figure 2 shows how all the holes in Figure 1 have been filled to complete the graph of the function f(x) = 2x, x E R. FIGURE 2 y = 2x, x real SECTION 1.4 Exponential Functions 47 The graphs of members of the family of functions y = bx are shown in Figure 3 for various values of the base b. Notice that all of these graphs pass through the same point (0, 1) because b0 = 1 for b = 0. Notice also that as the base b gets larger, the exponential function grows more rapidly (for x > 0). If 0 < b < 1, then bx approaches 0 as x becomes large. If b > 1, then bx approaches 0 as x decreases through negative values. In both cases the x-axis is a horizontal asymptote. These matters are discussed in Sec tion 2.6. 5® FIGURE 3 You can see from Figure 3 that there are basically three kinds of exponential functions y = bx. If 0 < b < 1, the exponential function decreases; if b = 1, it is a constant; and if b > 1, it increases. These three cases are illustrated in Figure 4. Observe that if b = 1, then the exponential function y = bx has domain R and range (0, -). Notice also that, since (1/b)x = 1/bx = b—x, the graph of y = (1/b)x is just the reflection of the graph of y = bx about the y-axis. \ y y y / (0, 1) 1 . . (0, 1) 0 x 0 x . x 0 FIGURE 4 (a) y=b®, 01 One reason for the importance of the exponential function lies in the following properties. If x and y are rational numbers, then these laws are well known from elementary algebra. It can be proved that they remain true for arbitrary real numbers x and y. www.stewartcalculus.com For review and practice using the Laws of Exponents, click on Review of Algebra. If a and b are positive numbers and x and y are any real numbers, then bx 1. bx+y = bxby 2. bx—y = by 3. (bx)y = bxy 4. (ab)x = axbx EXAMPLE 1 Sketch the graph of the function y = 3 — 2x and determine its domain and range. For a review of reflecting and shifting SOLUTION First we reflect the graph of y = 2x [shown in Figures 2 and 5(a)] about the graphs, see Section 1.3. x-axis to get the graph of y = —2x in Figure 5(b). Then we shift the graph of y = —2x 48 CHAPTER 1 Functions and Models upward 3 units to obtain the graph of y = 3 — 2x in Figure 5(c). The domain is R and the range is (--, 3). FIGURE 5~ (a) y=2x 0 _1 \ (b) y=_2x (c) y=3-2x ^ EXAMPLE 2 Use a graphing device to compare the exponential function f(x) = 2x and the power function g(x) = x2. Which function grows more quickly when x is large? SOLUTION Figure 6 shows both functions graphed in the viewing rectangle [-2, 6] by [0, 40]. We see that the graphs intersect three times, but for x > 4 the graph of f(x) = 2x stays above the graph of g(x) = x2. Figure 7 gives a more global view and shows that for large values of x, the exponential function y = 2x grows far more rapidly than the power function y = x2. 40 y=xz 6 _2 FIGURE 6~ ^ Example 2 shows that y = 2x increases more quickly than y = x2. To demonstrate just how quickly f (x) = 2x increases, let’s perform the following thought experiment. Suppose we start with a piece of paper a thousandth of an inch thick and we fold it in half 50 times. Each time we fold the paper in half, the thickness of the paper doubles, so the thickness of the resulting paper would be 250/1000 inches. How thick do you think that is? It works out to be more than 17 million miles! 250 y=2x y=xz 0 — 8 FIGURE 7~ ^ Applications of Exponential Functions The exponential function occurs very frequently in mathematical models of nature and society. Here we indicate briefly how it arises in the description of population growth and radioactive decay. In later chapters we will pursue these and other applications in greater detail. First we consider a population of bacteria in a homogeneous nutrient medium. Suppose that by sampling the population at certain intervals it is determined that the population doubles every hour. If the number of bacteria at time t is p(t), where t is measured in hours, and the initial population is p(0) = 1000, then we have p(1) = 2p(0) = 2 X 1000 p(2) = 2p(1) = 22 X 1000 p(3) = 2p(2) = 23 X 1000