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 11: 4

Answer

$1.\ \ \mathrm{Domain:} \ \ (-\infty,0] \cup [3,\infty)$ $2.\ \ \mathrm{Range:} \ \ [0,\infty)$

Work Step by Step

$1.$ We can take the square root only for positive numbers. $x^2-3x>=0$, $\ \ $ solve for $x$. $x(x-3)>=0$ The product of two numbers will be positive provided either both are negative or both are positive. $\mathrm{I}.$ When both are negative: $x<=0 \ \ and \ \ x-3<=-0$ All numbers that are $\ <=0\ $ satisfy both inequalities. So, $\ \ x\in(-\infty,0]$ $\mathrm{II}.$ When both are positive: $x>=0 \ \ and \ \ x-3>=0$ All numbers that are $\ >=3\ $ satisfy both inequalities. So, $\ \ x\in[3,\infty)$ Therefore, domain of the function is, $\ \ (-\infty,0] \cup [3,\infty)$ $2.$ Range of function is $\ \ [0,\infty]\ \ $ because the square root from any positive number is positive.
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