230 CHAPTER 2 Graphs and Functions –2 1 2 4 6 5 x y 0 y is increasing. y is decreasing. y is constant. f Figure 27 Increasing, Decreasing, and Constant Functions Suppose that a function ƒ is defined over an open interval I and x1 and x2 are in I. (a) ƒ increases over I if, whenever x1 6x2, ƒ1x12 6ƒ1x22. (b) ƒ decreases over I if, whenever x1 6x2, ƒ1x12 7ƒ1x22. (c) ƒ is constant over I if, for every x1 and x2, ƒ1x12 = ƒ1x22. Figure 28 illustrates these ideas. x y f (x1) f (x2) x1 x2 0 Whenever x1 < x2, and f (x1) < f (x2), f is increasing. y = f(x) (a) y x f (x2) f (x1) x1 x2 0 Whenever x1 < x2, and f (x1) > f (x2), f is decreasing. y = f(x) (b) y x f(x1) = f(x2) x1 x2 0 For every x1and x2, if f (x1) = f (x2), then f is constant. y = f(x) (c) Figure 28 Now find ƒ1-22 and ƒ1p2. ƒ1x2 = 1 4 x - 5 4 ƒ1-22 = 1 4 1-22 - 5 4 Let x = -2. ƒ1-22 = - 7 4 ƒ1p2 = 1 4 p - 5 4 Let x = p. S Now Try Exercises 77 and 81. Increasing, Decreasing, and Constant Functions Informally speaking, a function increases over an open interval of its domain if its graph rises from left to right on the interval. It decreases over an open interval of its domain if its graph falls from left to right on the interval. It is constant over an open interval of its domain if its graph is horizontal on the interval. For example, consider Figure 27. • The function increases over the open interval 1-2, 12 because the y-values continue to get larger for x-values in that interval. • The function is constant over the open interval 11, 42 because the y-values are always 5 for all x-values there. • The function decreases over the open interval 14, 62 because in that interval the y-values continuously get smaller. The intervals refer to the x-values where the y-values increase, decrease, or are constant. The formal definitions of these concepts follow.
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