By John Mace
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1 SOLUTION We start with the graph of y e x from Figures 13 and 15(a) and reflect about the y-axis to get the graph of y eϪx in Figure 15(b). (Notice that the graph crosses the y-axis with a slope of Ϫ1). Then we compress the graph vertically by a factor of 2 to obtain the graph of y 12 eϪx in Figure 15(c). Finally, we shift the graph downward one unit to get the desired graph in Figure 15(d). The domain is ޒand the range is ͑Ϫ1, ϱ͒. y 1 0 y y y 1 1 1 0 x 0 x 0 x x y=_1 (a) y=´ (d) y= 21 e–®-1 (c) y= 21 e–® (b) y=e–® FIGURE 15 How far to the right do you think we would have to go for the height of the graph of y e x to exceed a million?
34. The curves with equations 21–23 |||| Find all solutions of the equation correct to two decimal places. 21. x 3 Ϫ 9x 2 Ϫ 4 0 y 22. x 3 4x Ϫ 1 23. x 2 sin x ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 24. We saw in Example 9 that the equation cos x x has exactly one solution. 3x has three solutions and find their values correct to two decimal places. (b) Find an approximate value of m such that the equation cos x mx has exactly two solutions. 25. Use graphs to determine which of the functions f ͑x͒ 10x 2 and t͑x͒ x 3͞10 is eventually larger (that is, larger when x is very large).
When does the graph of t finally surpass the graph of f ? e x Ͼ 1,000,000,000. 25. Under ideal conditions a certain bacteria population is known 1 1 Ϫ ex (b) t͑t͒ s1 Ϫ 2 t ■ ■ ; 24. Use a graph to estimate the values of x such that (b) f ͑x͒ 16. (a) t͑t͒ sin͑eϪt ͒ ■ 10 x ; 23. Compare the functions f ͑x͒ x and t͑x͒ e by graphing Find the domain of each function. 15. (a) f ͑x͒ ■ both functions in several viewing rectangles. Find all points of intersection of the graphs correct to one decimal place.
Beginner's Arabic Script by John Mace