SECTION 5.3 Exponential Functions 303 Table 5 illustrates what happens to the defining expression (2) as n takes on increasingly large values.The last number in the right column in the table approximates e correct to nine decimal places. That is, = e 2.718281828 .… Remember, the three dots indicate that the decimal places continue. Because these decimal places continue but do not repeat, e is an irrational number. The number e is often expressed as a decimal rounded to a specific number of places. For example, ≈ e 2.71828 is rounded to five decimal places. n n 1 n 1 1 + n 1 1 n ( ) + 1 1 2 2 2 0.5 1.5 2.25 5 0.2 1.2 2.48832 10 0.1 1.1 2.59374246 100 0.01 1.01 2.704813829 1,000 0.001 1.001 2.716923932 10,000 0.0001 1.0001 2.718145927 100,000 0.00001 1.00001 2.718268237 1,000,000 0.000001 1.000001 2.718280469 10,000,000,000 − 10 10 + − 1 10 10 2.718281828 Table 5 The exponential function ( ) = f x e ,x whose base is the number e, occurs with such frequency in applications that it is usually referred to as the exponential function. Most calculators have the key ex or ( )x exp , which may be used to approximate the exponential function for a given value of x. Use your calculator to approximate ex for = − = − = = x x x x 2, 1, 0, 1, and = x 2. See Table 6 using a TI-84 Plus CE.The graph of the exponential function ( ) = f x ex is given in Figure 30(a). Since < < e 2 3, the graph of = y ex lies between the graphs of = y 2x and = y 3 .x Do you see why? See Figure 30(c) using a TI-84 Plus CE. Table 6 Figure 30 = y ex 10 Y1 5 ex 21 24 4 (b) (a) (c) 10 Y3 5 3 x Y2 5 2 x Y1 5 ex 21 24 4 y 5 0 (1, e) x y 3 6 3 (2, e2) (0, 1) (–1, ) (–2, ) e– 1 – 1 e2 Graphing an Exponential Function Using Transformations Graph ( ) = − − f x ex 3 and determine the domain, range, horizontal asymptote, and y-intercept of f. EXAMPLE 6 (continued)
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