Chapter 3.3 Practice

Questions

  1. For each rational function \(f\), use the asymptotic behavior and its behavior near its zeros and poles to sketch \(f.\)
    1. \(f(x)=\frac{3x-1}{x^2+x}\)
    2. \(f(x)=\frac{(x+4)(x-5)^3(x-9)^2}{5x^2(x^2-1)(x+5)^2}\)
    3. \(f(x)=\frac{5x+4}{x^2+6}\)
    4. \(f(x)=\frac{x^4+1}{x+1}\)
  2. For each polynomial \(f\), solve each inequality separately: \(f(x)\geq 0\), \(f(x)>0\), \(f(x)<0\) and \(f(x)\leq 0\).
    1. \(f(x)=2(x+3)^2(x+2)^3(x-5)^3(x-6)^8\)
    2. \(f(x)=5x^3(x+5)(x+1)^6(2x-3)^6(x-4)^7\)
    3. \(f(x)=4(x+5)^4(5x+2)^3(5-2x)^5(x-6)\)
    4. \(f(x)=-5x(x+5)^4(5x+2)^2(2x-5)(x-6)^3\)
  3. For each rational function \(f\), solve each inequality separately: \(f(x)\geq 0\), \(f(x)>0\), \(f(x)<0\) and \(f(x)\leq 0\).
    1. \(f(x)=\frac{1}{x^2(x-4)}\)
    2. \(f(x)=\frac{1}{-5(x+3)(x+1)(x-5)^2}\)
    3. \(f(x)=\frac{1}{-3x(x+3)(x+1)^2(x-3)}\)
    4. \(f(x)=\frac{3x-1}{x^2+x}\)
    5. \(f(x)=\frac{(x+4)(x-5)^3(x-9)^2}{5x^2(x^2-1)(x+5)^2}\)
    6. \(f(x)=\frac{5x+4}{x^2+6}\)
    7. \(f(x)=\frac{x^4+1}{x+1}\)

Answers

    1. solution to \(f(x)\geq 0\) is \((-\infty,-2]\cup[5,\infty)\);
      solution to \(f(x)>0\) is \((-\infty,-3)\cup(-3,-2)\cup(5,6)\cup(6,\infty)\);
      solution to \(f(x)<0\) is \((-2,5)\);
      solution to \(f(x)\leq 0\) is \(\{-3\}\cup[-2,5]\cup\{6\}.\)
    2. solution to \(f(x)\geq 0\) is \([-5,0]\cup\{1.5\}\cup[4,\infty)\);
      solution to \(f(x)>0\) is \((-5,-1)\cup(-1,0)\cup(4,\infty)\);
      solution to \(f(x)<0\) is \((-\infty,-5)\cup(0,-1.5)\cup(-1.5,4)\);
      solution to \(f(x)\leq 0\) is \((-\infty,-5]\cup\{-1\}\cup[0,4].\)
    3. solution to \(f(x)\geq 0\) is \(\{-5\}\cup[-0.4,2.5]\cup[6,\infty)\);
      solution to \(f(x)>0\) is \((-0.4,2.5)\cup(6,\infty)\);
      solution to \(f(x)<0\) is \((-\infty,-5)\cup(2.5,6)\);
      solution to \(f(x)\leq 0\) is \((-\infty,-5]\cup\{-0.4\}\cup[2.5,6].\)
    4. solution to \(f(x)\geq 0\) is \((-\infty,0]\cup[2.5,6]\);
      solution to \(f(x)>0\) is \((-\infty,-5)\cup(-5,-0.4)\cup(-0.4,0)\cup(2.5,6)\);
      solution to \(f(x)<0\) is \((0,2.5)\cup(6,\infty)\);
      solution to \(f(x)\leq 0\) is \(\{-5\}\cup\{-0.4\}\cup[0,2.5]\cup[6,\infty).\)
  3. For each rational function \(f\), solve each inequality separately: \(f(x)\geq 0\), \(f(x)>0\), \(f(x)<0\) and \(f(x)\leq 0\).
    1. solution to \(f(x)\geq 0\) is \((4,\infty)\);
      solution to \(f(x)>0\) is \((4,\infty)\);
      solution to \(f(x)<0\) is \((-\infty,0)\cup(0,4)\);
      solution to \(f(x)\leq 0\) is \((-\infty,0)\cup(0,4).\)
    2. solution to \(f(x)\geq 0\) is \((-3,-1)\);
      solution to \(f(x)>0\) is \((-3,-1)\);
      solution to \(f(x)<0\) is \((-\infty,-3)\cup(-1,5)\cup(5,\infty)\);
      solution to \(f(x)\leq 0\) is \((-\infty,-3)\cup(-1,5)\cup(5,\infty).\)
    3. solution to \(f(x)\geq 0\) is \((-3,-1)\cup(-1,0)\cup(3,\infty)\);
      solution to \(f(x)>0\) is \((-3,-1)\cup(-1,0)\cup(3,\infty)\);
      solution to \(f(x)<0\) is \((-\infty,-3)\cup(0,3)\);
      solution to \(f(x)\leq 0\) is \((-\infty,-3)\cup(0,3).\)
    4. solution to \(f(x)\geq 0\) is \((-1,0)\cup\left[\frac{1}{3},\infty\right)\);
      solution to \(f(x)>0\) is \((-1,0)\cup\left(\frac{1}{3},\infty\right)\);
      solution to \(f(x)<0\) is \((-\infty,-1)\cup\left(0,\frac{1}{3}\right)\);
      solution to \(f(x)\leq 0\) is \((-\infty,-1)\cup\left(0,\frac{1}{3}\right].\)
    5. solution to \(f(x)\geq 0\) is \((-\infty,-5)\cup(-5,-4]\cup(-1,0)\cup(0,1)\cup[5,\infty)\);
      solution to \(f(x)>0\) is \((-\infty,-5)\cup(-5,-4)\cup(-1,0)\cup(0,1)\cup(5,9)\cup(9,\infty)\);
      solution to \(f(x)<0\) is \((-4,-1)\cup(1,5)\);
      solution to \(f(x)\leq 0\) is \([-4,-1)\cup(1,5].\)
    6. solution to \(f(x)\geq 0\) is \(\left[-\frac{4}{5},\infty\right)\);
      solution to \(f(x)>0\) is \(\left(-\frac{4}{5},\infty\right)\);
      solution to \(f(x)<0\) is \(\left(-\infty,-\frac{4}{5}\right)\);
      solution to \(f(x)\leq 0\) is \(\left(-\infty,-\frac{4}{5}\right].\)
    7. solution to \(f(x)\geq 0\) is \((-1,\infty)\);
      solution to \(f(x)>0\) is \((-1,\infty)\);
      solution to \(f(x)<0\) is \((-\infty,-1)\);
      solution to \(f(x)\leq 0\) is \((-\infty,-1).\)

Return

© Copyright 2025 by the POC Writing Team: Bryan Carrillo, Yat Sun Poon, and David Weisbart. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the POC Writing Team.