SECTION 5.3 Exponential Functions 299 Graphs that look like the one in Figure 19 occur very frequently in a variety of situations. For example, the graph in Figure 20 shows the annual revenue of Amazon, Inc. from 2000 to 2023. One might conclude from this graph that Amazon’s annual revenue is growing exponentially. Later in this chapter, we discuss other situations that exhibit exponential growth. For now, we continue to explore properties of exponential functions. Using all this information, plot some of the points from Table 3 and connect them with a smooth, continuous curve, as shown in Figure 19. From the graph, we conclude that the range of f is ( )∞ 0, . We also conclude that f is an increasing function, and so f is one-to-one. 10 21 24 Y1 5 2 x 4 Figure 19 ( ) = f x 2x x y 3 (2, 4) (1, 2) (0, 1) ( ) –2, 6 3 1 – 4 ( ) –1, 1 – 2 ( ) –3, 1 – 8 y 5 0 Source: Amazon, Inc. Figure 20 2000 2001 2002 2003 2004 2005 2006 2008 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 Year 2009 2010 2007 Amazon Revenue Total Revenue ($ billions) 0 50 100 150 200 250 300 350 400 450 500 550 600 2022 2023 2.76 3.12 3.93 5.26 6.92 8.49 10.71 14.84 19.17 24.51 34.20 48.08 61.09 74.45 88.99 107.01 135.99 177.87 232.89 280.52 386.06 469.82 514.01 574.79 The graph of ( ) = f x 2x in Figure 19 is typical of all exponential functions of the form ( ) = f x ax with > a 1. Such functions are increasing functions and so are one-to-one.Their graphs lie above the x-axis, contain the point ( ) 0, 1 , and rise rapidly as →∞ x . As →−∞ x , the x-axis ( ) = y 0 is a horizontal asymptote. There are no vertical asymptotes. Finally, the graphs are smooth and continuous with no corners or gaps.
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