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: 46

Answer

$\mathrm{Decreasing:}\:\:\:(-\infty,0)$ $\mathrm{Increasing:}\:\:\:(0,\infty)$ Graph is symmetric with respect to the $\ \mathrm{y-axis}.$
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Work Step by Step

$\mathrm{First\:Part:}\:\:$ According to the definitions: A function $\ f\ $ defined on an interval is increasing on $\ (a, b)\ $ if for every $\ x_1, x_2\ $ $\in$ $(a, b)$ $\ x_1\le x_2\ $ implies that $\ f(x_1)\le f(x_2).\ $ A function $\ f\ $ defined on an interval is decreasing on $\ (a, b)\ $ if for every $\ x_1, x_2\ $ $\in$ $(a, b)$ $\ x_1\le x_2\ $ implies that $\ f(x_1)\ge f(x_2).\ $ We can write the given function $\ y=(-x)^{\frac{2}{3}}\ $ as $\ y=\sqrt[3] {(-x)^2}=\sqrt[3] {x^2}.$ First of all, create a table with a few points to sketch the graph. $\quad \mathrm{See\:the\:table\:and\:graph\:above.}$ $\mathrm{Second\:Part:}\:\:$ Graph is symmetric with respect to the $\ \mathrm{y-axis}.$ $\mathrm{Third\:Part:}\:\:$ The graph of the given function $\ y=\sqrt[3] {x^2}\ $ is decreasing on $\ (-\infty,0)\ $ and is increasing on $\ (0,\infty).$
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