272 CHAPTER 2 Graphs and Functions Another piecewise-defined function is the greatest integer function. (b) First graph ƒ1x2 = 2x + 3 for x … 0. Then for x 70, graph ƒ1x2 = -x2 + 3. The two graphs meet at the point 10, 32. See Figure 64. 3 5 1 (0, 3) 1 3 5 –3 –1 0 x y 2x + 3 if x " 0 –x2 + 3 if x + 0 f(x) = Figure 64 (b) Use the procedure described in part (a). The expression for the function is shown at the top of the screen in Figure 65. −10 −5 5 5 Figure 65 S Now Try Exercises 23 and 29. LOOKING AHEAD TO CALCULUS The greatest integer function is used in calculus as a classic example of how the limit of a function may not exist at a particular value in its domain. For a limit to exist, the left- and right-hand limits must be equal. We can see from the graph of the greatest integer function that for an integer value such as 3, as x approaches 3 from the left, function values are all 2, while as x approaches 3 from the right, function values are all 3. Because the left- and right-hand limits are different, the limit as x approaches 3 does not exist. ƒ1x2 = Œxœ The greatest integer function ƒ1x2 = Œxœ pairs every real number x with the greatest integer less than or equal to x. For example, Œ8.4œ = 8, Œ -5œ = -5, Œpœ = 3, and Œ -6.9œ = -7. The graph of ƒ1x2 = Œxœ is shown in Figure 66. In general, if ƒ1x2 = Œxœ, then for -2 … x 6 -1, ƒ1x2 = -2, for -1 … x 60, ƒ1x2 = -1, for 0 … x 61, ƒ1x2 = 0, for 1 … x 62, ƒ1x2 = 1, for 2 … x 63, ƒ1x2 = 2, and so on. Greatest Integer Function ƒ1x2 = Œxœ Domain: 1-∞, ∞2 Range: 5y y is an integer6 = 5c , -3, -2, -1, 0, 1, 2, 3, c6 x y -2 -2 -1.5 -2 -0.99 -1 0 0 0.001 0 3 3 3.99 3 • ƒ1x2 = Œxœ is constant on the open intervals c , 1-2, -12, 1-1, 02, 10, 12, 11, 22, 12, 32, c . • It is discontinuous at all integer values in its domain, 1-∞, ∞2. −5 −3 5 4 f(x) = [[x]] 3 2 4 1 –3 –4 4 –4 x 1 2 3 –3 0 –2 f(x) = [[x]] y The dots indicate that the graph continues indefinitely in the same pattern. Figure 66
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