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76.
Chapter 2 Graphs and Functions
84. False. For example, f ( x) x 3 is odd, and (2, 8) is on the graph but (–2, 8) is not.
2 f ( x) x 3 if x 2 x 4 if x 2
Graph the curve y x 2 3 to the left of x = 2, and graph the line y = –x + 4 to the right of x = 2. The graph has an open circle at (2, 7) and a closed circle at (2, 2).
77.
x if x 3 6 x if x 3 Draw the graph of y x to the left of x = 3, but do not include the endpoint. Draw the graph of y = 6 – x to the right of x = 3, including the endpoint. Observe that the endpoints of the two pieces coincide. f ( x)
78. Because x represents an integer, x x. Therefore, x x x x 2 x. 79. True. The graph of an even function is symmetric with respect to the y-axis. 80. True. The graph of a nonzero function cannot be symmetric with respect to the x-axis. Such a graph would fail the vertical line test 81. False. For example, f ( x) x 2 is even and (2, 4) is on the graph but (2, – 4) is not. 82. True. The graph of an odd function is symmetric with respect to the origin. 83. True. The constant function f x 0 is both
85. x y 2 10 10. Replace x with –x to obtain ( x) y 2 The result is not the same as the original equation, so the graph is not symmetric with respect to the y-axis. Replace y with –y to obtain x ( y) 2 10 x y 2 10. The result is the same as the original equation, so the graph is symmetric with respect to the x-axis. Replace x with –x and y with –y to obtain ( x) ( y) 2 10 ( x) y 2 10. The result is not the same as the original equation, so the graph is not symmetric with respect to the origin. The graph is symmetric with respect to the x-axis only. 86. 5 y 2 5x 2 30 Replace x with –x to obtain 5 y 2 5( x) 2 30 5 y 2 5x 2 30. The result is the same as the original equation, so the graph is symmetric with respect to the y-axis. Replace y with –y to obtain 5( y) 2 5x 2 30 5 y 2 5x 2 30. The result is the same as the original equation, so the graph is symmetric with respect to the x-axis. The graph is symmetric with respect to the y-axis and x-axis, so it must also be symmetric with respect to the origin. Note that this equation is the same as y 2 x 2 6 , which is a circle centered at the origin. 87. x 2 y 3 Replace x with –x to obtain ( x) 2 y 3 x 2 y 3 . The result is the same as the original equation, so the graph is symmetric with respect to the y-axis. Replace y with –y to obtain x 2 ( y)3 x 2 y 3 . The result is not the same as the original equation, so the graph is not symmetric with respect to the x-axis. Replace x with –x and y with –y to obtain ( x) 2 ( y)3 x 2 y 3 . The result is not the same as the original equation, so the graph is not symmetric with respect to the origin. Therefore, the graph is symmetric with respect to the y-axis only.
even and odd. Because f x 0 f x , the function is even. Also f x 0 0 f x , so the function is odd.
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