Precalc0105to0107-page36

Precalc0105to0107-page36 - g 1 ( ) = g 1 ( ) or g 1 ( ) = g...

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(Section 1.7: Symmetry Revisited) 1.7.6 Example 5 (A Polynomial Function that is Neither Even nor Odd) Let gt () = t 3 + t + 1 . Is g even, odd, or neither? Justify your answer. § Solution 1 (Short Cut) gt () is polynomial and has terms of odd degree (3 and 1) and a term of even degree (0). Therefore, g is neither even nor odd. § § Solution 2 (Rigorous) Dom g () = ± . Evaluate g ± t () and compare it to gt () and ± gt () . g ± t () = ± t () 3 + ± t () + 1 = ± t 3 ± t + 1 g is not even , because g ± t () is not equivalent to gt () on ± . g is not odd , because g ± t () is not equivalent to ± gt () on ± . • A counterexample suffices to justify both statements. For instance, g ± 1 () = ± 1 , and g 1 () = 3 . Then, g ± 1 () ² g 1 () , and g ± 1 () ²± g 1 () . WARNING 3 : If it had been true that
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Unformatted text preview: g 1 ( ) = g 1 ( ) or g 1 ( ) = g 1 ( ) , that would not have been enough to prove that g was even or odd. Examples cannot prove that a function g is even or odd, unless Dom g ( ) is a finite set. g is neither even nor odd. Let f be a (nontrivial) even function, and let g be a (nontrivial) odd function. Then, f + g and f g must be neither even nor odd. (Experiment with graphs and numbers. See the Exercises.) Example 6 (Revisiting Example 5) Let g t ( ) = t 3 + t + 1 ; this is not g in the box above. Since g t ( ) = 1 ( ) + t 3 + t ( ) , g can take the form even ( ) + odd ( ) , and thus g is neither even nor odd....
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This document was uploaded on 12/29/2011.

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