Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 2 - Section 2.6 - Transformations of Functions - 2.6 Exercises - Page 210: 103

Answer

The product of two even functions is an even function. The product of two odd functions is an even function. The product of an even function and an odd function is an odd function.

Work Step by Step

Suppose both $f$ and $g$ are even functions. We can find an expression for $f(-x)~g(-x)$: $f(-x)~g(-x) = f(x)~g(x)$ So the product of two even functions is an even function. Suppose both $f$ and $g$ are odd functions. We can find an expression for $f(-x)~g(-x)$: $f(-x)~g(-x) = [-f(x)]~[-g(x)] = f(x)~g(x)$ So the product of two odd functions is an even function. Suppose $f$ is an even function and $g$ is an odd function. We can find an expression for $f(-x)~g(-x)$: $f(-x)~g(-x) = [f(x)]~[-g(x)] = -f(x)~g(x)$ So the product of an even function and an odd function is an odd function.
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