273 2.6 Graphs of Basic Functions x y 1 2 3 4 1 2 3 4 –4–3–2 –2 –1 –3 –4 0 y = [[ x + 1]] 1 2 Figure 67 The greatest integer function is an example of a step function, a function with a graph that looks like a series of steps. EXAMPLE 3 Graphing a Greatest Integer Function Graph ƒ1x2 = fi 1 2 x + 1fl . SOLUTION If x is in the interval 30, 22, then y = 1. For x in 32, 42, y = 2, and so on. Some sample ordered pairs are given in the table. x 0 1 2 1 3 2 2 3 4 -1 -2 -3 y 1 1 1 1 2 2 3 0 0 -1 These ordered pairs suggest the graph shown in Figure 67. The domain is 1-∞, ∞2. The range is 5c , -2, -1, 0, 1, 2, c6. S Now Try Exercise 55. EXAMPLE 4 Applying the Greatest Integer Function Concept An express mail company charges $25 for a package weighing up to 2 lb. For each additional pound or fraction of a pound, there is an additional charge of $3. Let y = D1x2 represent the cost to send a package weighing x pounds. Graph y = D1x2 for x in the interval 10, 64. SOLUTION For x in the interval 10, 24, we obtain y = 25. For x in 12, 34, y = 25 + 3 = 28. For x in 13, 44, y = 28 + 3 = 31, and so on. The graph, which is that of a step function, is shown in Figure 68. In this case, the first step has a different length. S Now Try Exercise 59. x y 0 1 2 3 4 5 6 20 30 40 Pounds Dollars y = D(x) Figure 68 NOTE A piecewise-defined function representing the graph in Figure 68 is D1x2 = e 25 if 0 6x … 2 25 - 3Œ2 - xœ if x 72. Verify that this function is valid by substituting the x-values 2, 2.5, 3, 3.5, and 4. The Relation x =y 2 Recall that a function is a relation where every domain value is paired with one and only one range value. Consider the relation defined by the equation x =y2, which is not a function. Notice from the table of selected ordered pairs on the next page that this relation has two different y-values for each positive value of x. If we plot the points from the table and join them with a smooth curve, we find that the graph of x = y2 is a parabola opening to the right with vertex 10, 02. See Figure 69(a) on the next page. The domain is 30, ∞2 and the range is 1-∞, ∞2. To use a calculator in function mode to graph the relation x = y2, we graph the two functions y1 = 2x (to generate the top half of the parabola) and y2 = -2x (to generate the bottom half). See Figure 69(b) on the next page. 7
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