University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 1 - Section 1.1 - Functions and Their Graphs - Exercises - Page 12: 56



Work Step by Step

$\mathrm{Function\:Parity\:Definition:} $ $\mathrm{Even\:Function:}\:\: $ A function is even if $\ h(-t)=h(t)\ $ for all $\ x\in \mathbb{R}. $ $\mathrm{Odd\:Function:}\:\: $ A function is odd if $\ h(-t)=-h(t)\ $ for all $\ x\in \mathbb{R}. $ $h(t)=|t^3|$ $h(-t)=|(-t)^3|$ $h(-t)=|-t^3|$ By applying the absolute rule $\ |-a|\:=\:|a|\:\ $, we have: $h(-t)=|t^3|$ Now, $-h(t)=-|t^3|$ Since, $h(-t)= h(t)\mathrm{,\:therefore\:}|t^3|\mathrm{\:is\:an\:even\:function}$ $h(-t)\ne -h(t)\mathrm{,\:therefore\:}|t^3|\mathrm{\:is\:not\:an\:odd\:function}$
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