Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 1 - Precalculus Review - 1.1 Real Numbers, Functions, and Graphs - Exercises - Page 12: 80

Answer

(a) and (d) all even, (b) and (c) are odd.

Work Step by Step

We know that odd functions obey the principle: $f(-x)=-f(x)$ and even functions follow the principle: $f(-x)=f(x)$ (a) and (d) are even functions. This is because multiplying or dividing two odd functions makes them even (the negative sign cancels out). $f(-x)g(-x)=-f(x)*-g(x)=f(x)g(x)$ Similarly, division will also lead to a negative sign cancellation. (c) is an odd function. Consider: $f(-x)-g(-x)=-f(x)-(-g(x))=-f(x)+g(x)=-(f(x)-g(x))$ Hence, it is odd based on the principles stated above. (b) is an odd function. We can rewrite the function as follows: $f(x)^3= f(x) \times f(x)^2$ Then $f(-x)^3=f(-x)\times f(-x)^2=-f(x)\times f(x)^2=-f(x)^3$ Hence, it is odd based on the principles stated above.
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