88 CHAPTER 2 Functions and Their Graphs Refer to page 22, where the tests for symmetry are listed.The results below follow. THEOREM Graphs of Even and Odd Functions • A function is even if and only if its graph is symmetric with respect to the y -axis. • A function is odd if and only if its graph is symmetric with respect to the origin. Solution (a) The graph in Figure 22(a) is that of an even function, because the graph is symmetric with respect to the y -axis. (b) The function whose graph is given in Figure 22(b) is neither even nor odd, because the graph is neither symmetric with respect to the y -axis nor symmetric with respect to the origin. (c) The function whose graph is given in Figure 22(c) is odd, because its graph is symmetric with respect to the origin. Identifying Even and Odd Functions from a Graph Determine whether each graph given in Figure 22 is the graph of an even function, an odd function, or a function that is neither even nor odd. EXAMPLE 1 Figure 22 x y (a) x y (c) x y (b) Now Work PROBLEMS 25(a), (b) , AND (d) 2 Identify Even and Odd Functions from an Equation A graphing utility can be used to conjecture whether a function is even, odd, or neither. Remember that when the graph of an even function contains the point x y , , ( ) it must also contain the point x y , . ( ) − Therefore, if the graph shows evidence of symmetry with respect to the y -axis, we would conjecture that the function is even. In addition, if the graph shows evidence of symmetry with respect to the origin, we would conjecture that the function is odd. Identifying Even and Odd Functions Use a graphing utility to conjecture whether each of the following functions is even, odd, or neither. Then algebraically determine whether the graph is symmetric with respect to the y -axis or with respect to the origin. (a) f x x 5 2 ( ) = − (b) g x x 1 3 ( ) = − (c) h x x x 5 3 ( ) = − EXAMPLE 2 Figure 23 10 25 24 4 Solution (a) Graph the function. See Figure 23. It appears that the graph is symmetric with respect to the y -axis. We conjecture that the function is even. To algebraically verify the conjecture, replace x by x− in f x x 5. 2 ( ) = − Then f x x x f x 5 5 2 2 ( ) ( ) ( ) − = − − = − = Since f x f x , ( ) ( ) − = we conclude that f is an even function and that the graph is symmetric with respect to the y -axis.
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