Learn – Algebra/Precalculus – Rationals

Definition

Exercises

Problem set

For each of the following, determine if the given expression is rational.

  1. x^{-3}
  2. x^2+\frac{1}{x^2}
  3. \frac{x^{1/2}+1}{x^2+1}
  4. \frac{x^{-2}+1}{x^{-1}+1}
  5. \frac{2^x}{3^x+1}
  6. \frac{x^2+\sin x}{1+\cos x}

Graphing

Exercises

Problem set

For each of the following problems, graph f(x) and g(x) in a single coordinate plane.

  1. f(x) = \frac{1}{x},\,\, g(x) = \frac{1}{x^2}
  2. f(x) = \frac{1}{x^2},\,\, g(x) = \frac{1}{x^3}
  3. f(x) = \frac{1}{x},\,\, g(x) = \frac{1}{x^3}
  4. f(x) = -\frac{1}{x},\,\, g(x) = -\frac{1}{x^3}
  5. f(x) = -\frac{1}{x^2},\,\, g(x) = -\frac{1}{x^4}

Problem set

Determine the zeros, y-intercepts, vertical asymptotes and end behavior (horizontal asymptotes) of the following functions. Use those to draw their approximate graphs.

  1. \frac{(x-3)(x+5)}{(x+3)(x-7)(x+10)}
  2. \frac{(x-3)(x+1)}{(x-1)^2(3+x)(x-5)}
  3. \frac{(x+5)(x-1)}{(x-3)(1-3x)^2}
  4. \frac{(x+5)(5-x)}{(1-x)(7-2x)^2(x+1)^2}
  5. \frac{(-x-5)(9-2x)^2}{9(1-2x)(3-x)^2(x+1)^2}
  6. -\frac{(2x+11)^2(5-x)^3}{11x(3-x)^3(2x+5)^2}

Problem set

Determine the zeros, y-intercepts, vertical asymptotes, holes and end behavior (horizontal asymptotes) of the following functions. Use those to draw their approximate graphs.

  1. \frac{(x-3)(x+2)}{(x-3)(x+5)(x-5)}
  2. \frac{(x+3)(1-x)}{(x-1)(x+6)(7-2x)}
  3. \frac{(x+3)(x-1)^2}{(x-1)(-x-5)^2(7-2x)}
  4. -\frac{(11x+2)(x-1)^2}{11(1-x)^2(x+3)(7-2x)}
  5. -\frac{(11x+2)(x-1)}{11(1-x)^2(x+3)(7-2x)}
  6. \frac{(11x+2)^2(x-1)(x-5)}{11(1-x)^3(x+2)(5-2x)}

Problem set

Graph the following functions.

  1. \frac{-3x^2+6x}{3x^2-7x+4}
  2. \frac{5x^2+12x-9}{2x^2+5x-3}
  3. \frac{6x^3-x^2-3}{6x^2-x-1}
  4. \frac{2x^3+2x^2-x}{4x^2+4x-3}
  5. \frac{2x^4+2x^3-\frac{3}{2}x^2+x-1}{4x^2+4x-3}
  6. \frac{x^4-\frac{1}{2}x^3-x^2-2}{2x^2-x-3}

Problem set

Graph the following functions.

  1. \frac{(x-1)(x+5)}{(x-3)(x+2)}
  2. \frac{(2-5x)(x+5)}{(x-5)(x+2)}
  3. \frac{(1-2x)(x-7)(4+x)}{(3-x)(-x-1)(x-5)}
  4. \frac{(2-5x)(x-7)(4+x)^2}{x(3+2x)^2}
  5. \frac{(1-x)^2(x-7)(4+x)^2}{x^3(x-5)}

Problem set

Graph the following functions.

  1. \frac{x(1-x)^2(x+6)}{(x+3)(x-3)}
  2. \frac{(x+1)^2(x-1)^2(x+6)}{(x+3)^2(3-x)}

Problem set

Determine the zeros, y-intercepts, vertical asymptotes and end behavior (horizontal asymptotes) of the following functions. Use those to draw their approximate graphs.

  1. \frac{(x-3)^2(x+5)}{(x+3)(x-5)(x-1)}
  2. \frac{(x-3)(x+5)}{x(x-7)(x+10)}
  3. \frac{(x+5)^2}{x(x-7)(x+10)}
  4. \frac{8x^3+27}{(8x^3-27)x}