Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 1 - Functions - 1.1 Review of Functions - 1.1 Exercises: 84

Answer

$a.$ Not necessary. $b.$ Necessary.

Work Step by Step

$a.$ Not necessarily. We need that $f(-x)=f(x)$ i.e. the graph of the function has to be symmetric with respect to the $y$ axis, but it can intersect it anywhere. The example is $f(x)=x^2+1$. This is an even function but $f(0)=1.$ $b.$ It is necessary. For all $x$ we have $f(-x)=-f(x)$ so for $x=0$ it must be that $f(-0)=-f(0)$ or simply $f(0)=-f(0)$ which is possible only if $f(0)=0.$
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