104 CHAPTER 2 Functions and Their Graphs Absolute Value Function f x x ( ) = See Figure 47. The domain of the absolute value function is the set of all real numbers; its range is the set of nonnegative real numbers.The intercept of the graph is at 0, 0 . ( ) If x 0, ≥ then f x x, ( ) = and the graph of f is part of the line y x; = if x 0, < then f x x, ( ) = − and the graph of f is part of the line y x. = − The absolute value function is an even function; it is decreasing on the interval , 0 ( ] −∞ and increasing on the interval 0, . [ )∞ Figure 47 Absolute Value Function x y 3 3 23 f(x) = ƒ x ƒ (1, 1) (0, 0) (21, 1) (2, 2) (22, 2) The notation x int( ) stands for the greatest integer less than or equal to x. For example, int 1 1 int 2.5 2 int 1 2 0 int 3 4 1 int 3 π ( ) ( ) ( ) ( ) ( ) = = = − = − = DEFINITION Greatest Integer Function f x x x int greatest integer less than or equal to ( ) ( ) = = * We obtain the graph of f x x int ( ) ( ) = by plotting several points. See Table 7. For values of x x , 1 0, − ≤ < the value of f x x int ( ) ( ) = is 1; − for values of x x , 0 1, ≤ < the value of f is 0. See Figure 48 for the graph. The domain of the greatest integer function is the set of all real numbers; its range is the set of integers. The y -intercept of the graph is 0. The x -intercepts lie in the interval 0, 1 . [ ) The greatest integer function is neither even nor odd. It is constant on every interval of the form k k , 1 , [ ) + for k an integer. In Figure 48, a solid dot indicates, for example, that at x 1 = the value of f is f 1 1; ( ) = an open circle is used to show that the value of f is not 0 at x 1. = Although a precise definition requires calculus, in a rough sense, a function is continuous if its graph has no gaps or holes and can be traced without lifting the pencil from the paper on which the graph is drawn. A function is discontinuous if its graph has gaps or holes and cannot be traced without lifting the pencil from the paper. The greatest integer function is also called a step function . At = x 0, = ± = ± x x 1, 2, and so on, this function is discontinuous because, at integer values, the graph suddenly “steps” from one value to another without taking on any of the intermediate values. For example, to the immediate left of x 3, = the y -coordinates of the points on the graph are 2, and at x 3 = and to the immediate right of x 3, = the y -coordinates of the points on the graph are 3. Figure 48 Greatest Integer Function x y 4 2 22 2 4 23 Figure 49 f x x int ( ) ( ) = 22 22 6 (a) TI-83 Plus, connected mode 6 22 22 6 6 (b) TI-83 Plus, dot mode (d) Desmos f(x) 5 floor(x) 6 −2 6 −2 (c) TI-84 Plus CE COMMENT When graphing a function using a graphing utility, typically you can choose either connected mode , in which points plotted on the screen are connected, making the graph appear continuous, or dot mode , in which only the points plotted appear. When graphing the greatest integer function with a graphing utility, it may be necessary to be in dot mode . This is to prevent the utility from “connecting the dots” when f x( ) changes from one integer value to the next. However, some utilities will display the gaps even when in “connected” mode. See Figure 49. ■ *Some texts use the notation ( ) [ ] = f x x or call the greatest integer function the “floor function” and use the notation ⎣ ⎦ ( ) = f x x . x ( ) ( ) = = y f x x int x, y ( ) −1 −1 ( ) − − 1, 1 − 1 2 −1 ( ) − − 1 2 , 1 − 1 4 −1 ( ) − − 1 4 , 1 0 0 ( ) 0, 0 1 4 0 ( ) 1 4 , 0 1 2 0 ( ) 1 2 , 0 3 4 0 ( ) 3 4 , 0 Table 7
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