Algebra and Trigonometry 10th Edition

Published by Cengage Learning
ISBN 10: 9781337271172
ISBN 13: 978-1-33727-117-2

Chapter 2 - 2.6 - Combinations of Functions: Composite Functions - 2.6 Exercises - Page 221: 69


$(fg)(-x)=(fg)(x)$ if both $f(x)$ and $g(x)$ are odd or if both $f(x)$ and $g(x)$ are even.

Work Step by Step

1) Let $f(x)$ and $g(x)$ be two odd functions. That is: $f(-x)=-f(x)$ and $g(-x)=-g(x)$ $(fg)(x)=f(x)g(x)$ So, $(fg)(-x)=f(-x)g(-x)=-f(x)[-g(x)]=f(x)g(x)=(fg)(x)$ 2) Let $f(x)$ and $g(x)$ be two even functions. That is: $f(-x)=f(x)$ and $g(-x)=g(x)$ $(fg)(x)=f(x)g(x)$ So, $(fg)(-x)=f(-x)g(-x)=f(x)g(x)=(fg)(x)$
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