Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 1 - Review - Concept Check: 3

Answer

(a) The function is even if for every $x$ from its domain $f(-x)=f(x)$. If the graph of the function is symmetric with respect to $y$ axis then the function is even. The examples are $f(x)=x^2$, $f(x)=\cos x$, $f(x)=2x^4+1.$ (b) (a) The function is odd if for every $x$ from its domain $f(-x)=-f(x)$. If the graph of the function is symmetric with respect to the origin then the function is even. The examples are $f(x)=x$, $f(x)=\sin x$, $f(x)=x^3.$

Work Step by Step

(a) The function is even if for every $x$ from its domain $f(-x)=f(x)$. If the graph of the function is symmetric with respect to $y$ axis then the function is even. This is because whatever "happens" to the function at $x$, the same will happen at $-x$. The examples are $f(x)=x^2$, $f(x)=\cos x$, $f(x)=2x^4+1.$ This is because $$(-x)^2=x^2,\quad \cos{-x}=\cos x,\quad 2(-x)^4+1=2x^4+1$$ (b) (a) The function is odd if for every $x$ from its domain $f(-x)=-f(x)$. If the graph of the function is symmetric with respect to the origin then the function is even. This is because if the function passes through the point $(x,y)$, it will pass through the point $(-x,-y)$. The examples are $f(x)=x$, $f(x)=\sin x$, $f(x)=x^3.$ This is because $$(-x)=-x,\quad (-x)^3=-x^3,\quad \sin(-x)=-\sin x.$$
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