Chapter 3.2 Practice

Questions

  1. For each rational function \(f\), determine its zeros and poles. Also list the order of each zero and pole.
    1. \(f(x)=\frac{1}{x^2(x-4)}\)
    2. \(f(x)=\frac{3x-1}{x^2+x}\)
    3. \(f(x)=\frac{(x+4)(x-5)^3(x-9)^2}{5x^2(x^2-1)(x+5)^2}\)
    4. \(f(x)=\frac{5x+4}{x^2+6}\)
    5. \(f(x)=\frac{x^4+1}{x+1}\)
  2. For each rational function \(f\), determine asymptotic behavior. If it has any, state its horizontal asymptote.
    1. \(f(x)=\frac{1}{x^2(x-4)}\)
    2. \(f(x)=\frac{3x-1}{x^2+x}\)
    3. \(f(x)=\frac{(x+4)(x-5)^3(x-9)^2}{5x^2(x^2-1)(x+5)^2}\)
    4. \(f(x)=\frac{5x+4}{x^2+6}\)
    5. \(f(x)=\frac{x^4+1}{x+1}\)
  3. For each polynomial function \(p\), take \(q\) to be the rational function \[q(x)=\frac{1}{p(x)}.\] Use \(y\)-axis inversion and a sketch of \(p\) to sketch \(q\).
    1. \(p(x)=x^2(x-4)\)
    2. \(p(x)=-5(x+3)(x+1)(x-5)^2\)
    3. \(p(x)=-3x(x+3)(x+1)^2(x-3)\)

Answers

    1. no zeros, poles at \(x=0\) and \(x=4\) with respective orders of \(2\) and \(1\)
    2. zero at \(x=\frac{1}{3}\) with order \(1\), poles at \(x=-1\) and \(x=0\) with respective orders of \(1\) and \(1\)
    3. zeros at \(x=-4\), \(x=5\) and \(x=9\) with respective orders of \(1\),\(3\), and \(2\), poles at \(x=-5\), \(x=-1\), \(x=0\), \(x=1\) with respective orders of \(2\),\(1\),\(2\) and \(1\)
    4. zero at \(x=-\frac{4}{5}\) with order \(1\), no pole
    5. no zeros, pole at \(x=-1\) with order \(1\)
    1. \(\frac{1}{x^3}\), horizontal asymptote at \(y=0\)
    2. \(\frac{3}{x}\), horizontal asymptote at \(y=0\)
    3. \(\frac{1}{5}\), horizontal asymptote \(y=\frac{1}{5}\)
    4. \(\frac{5}{x}\), horizontal asymptote at \(y=0\)
    5. \(x^3\), no horizontal asymptote

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