Chapter 3.3 Practice

\[ \definecolor{ucrblue}{rgb}{0.0627,0.3843,0.6039} \definecolor{ucrgold}{rgb}{0.9686,0.6941,0.2824} \definecolor{ucrred}{rgb}{0.8941,0,0.1686} \definecolor{ucrgreen}{rgb}{0.4706,0.7451,0.1255} \definecolor{ucraccent}{rgb}{1.0000,0.9569,0.8392} \DeclareMathOperator*{\LO}{O} \DeclareMathOperator*{\Lo}{o} \DeclareMathOperator*{\Recip}{Recip} \DeclareMathOperator*{\abs}{abs} \DeclareMathOperator{\pow}{pow} \]

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. 1. 
   2. 
   3. 
   4. 
2. 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 \((-\infty,-1).\)

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