Section 1.4C01S04.001:Becauseg(x) = 2xincreases—first slowly, then rapidly—on the set of all real numbers, withvalues in the range (0,+∞), the given functionf(x) = 2x−1 must increase in the same way, but withvalues in the range (−1,+∞). Therefore its graph is the one shown in Fig. 1.4.29.C01S04.002:Given:f(x) = 2−3−x. The graph ofg(x) = 3xincreases, first slowly, then rapidly, on itsdomain the setRof all real numbers. Henceh(x) = 3−xdecreases, first rapidly, then slowly, onR, withvalues in the interval (0,+∞). Hencej(x) =−3−xincreases, first rapidly, then slowly, onR, with valuesin the interval (−∞,0). Thereforef(x) = 2−3−xincreases, first rapidly, then slowly, onR, with values inthe interval (−∞,2). Therefore its graph must be the one shown in Fig. 1.4.33.C01S04.003:The graph off(x) = 1 + cosxis simply the graph of the ordinary cosine function raised 1unit—moved upward 1 unit in the positivey-direction. Hence its graph is the one shown in Fig. 1.4.27.
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