2. Rational Functions
  3. Vertical Asymptotes and Discontinuity

Exercises


  1. The rational function \( f(x) = \dfrac{2x}{x-4} \) has a vertical asymptote at \( x = 4 \).
    1. Support this statement using a table of values. Investigate the behaviour of the function as \( x \) approaches \( 4 \) from the left \( (x\rightarrow4^-) \) and as \( x \) approaches \( 4 \) from the right \( (x \rightarrow 4^+) \). What happens to the value of \( y \) in each situation?
    2. Support this statement algebraically by discussing the value of the numerator and denominator of the function at \( x = 4 \).
  2. The rational function \( f(x) = \dfrac{-3x-6}{2x+4} \) has a hole in its graph at \( x = -2 \).
    1. Support this statement using a table of values. Investigate the behaviour of the function as \( x \) approaches \( -2 \) from the left \( (x\rightarrow-2^-) \) and as \( x \) approaches \( -2 \) from the right \( (x \rightarrow -2^+) \). What happens to the value of \( y \) in each situation?
    2. Support this statement algebraically by discussing the value of the numerator and denominator of the function at \( x = -2 \).
    3. What are the exact coordinates of the hole?
  3. Consider the rational function \( f(x) = \dfrac{-3x^2-15x-18}{x^2+x-2} \).
    1. Explain why the function is discontinuous at \( x = -2 \) and \( x = 1 \).
    2. Using a table of values, investigate the behaviour of the function near \( x = -2 \) and \( x = 1 \).
    3. Based on your findings in part (b) describe the behaviour of the graph of the function at \( x = -2 \) and \( x = 1 \).
    4. How can the behaviour of the graph of \( f(x) \) at \( x=-2 \) and \( x=1 \) be determined algebraically using the equation?
  4. Given \( a \neq b, a, b \in \mathbb{R} \), discuss the similarities and differences, if any, in the discontinuities of the graphs of the following pairs of functions:
    1. \( f(x) = \dfrac{(x-a)(x-b)}{x-b} \) and \( g(x) = x-a \)
    2. \( f(x) = \dfrac{(x-a)(x-b)}{(x-b)^2} \) and \( g(x) = \dfrac{x-a}{x-b} \)
    3. \( f(x) = \dfrac{(x-b)^2}{(x-a)(x-b)} \) and \( g(x) = \dfrac{x-b}{x-a} \)
  5. State the domain of each function, then determine the equation of any vertical asymptotes and/or coordinates of any holes in the graph of the function.
    1. \( f(x) = \dfrac{2x}{x-3} \)
    2. \( f(x) = \dfrac{2x^2+x}{x^2-5x+6} \)
    3. \( f(x) = \dfrac{3x^2-21x}{6x^2-39x-21} \)
    4. \( f(x) = \dfrac{x^3+x}{6x^3+x^2-x} \)
    5. \( f(x) = \dfrac{x^3-x}{2x^3+9x^2+13x+6} \)
  6. For each rational function, identify any vertical asymptote(s) and discuss the behaviour of the function near these asymptote(s).
    1. \( f(x) = \dfrac{-x}{x-4} \)
    2. \( f(x) = \dfrac{-3x+1}{2x+5} \)
    3. \( f(x) = \dfrac{x^2}{x^2-9} \)
  7. Based on your understanding of vertical asymptotes and holes, choose the most appropriate graph for each equation below. Support your choice.
    1. \( f(x) = \dfrac{x-3}{x^2-3x} \)
    2. \( g(x) = \dfrac{x^2+3x}{x^2+9} \)
    3. \( h(x) = \dfrac{-3}{x+3} \)
    4. \( p(x) = \dfrac{3x}{x^2+3x} \)
    5. \( q(x) = \dfrac{x+3}{x^2-9} \)
    6. \( r(x) = \dfrac{3x+9}{x^2+6x+9} \)
    I.
    Vertical asymptote at x=3, hole at (-3, -1/3), positive for x>3, negative for x<3
    II.
    Vertical asymptote at x=0, hole at (3, 1/3), positive for x>0, negative for x<0
     
    III.
    Vertical asymptote at x=-3, hole at (0, 1), positive for x>3, negative for x<3
    IV.
    Vertical asymptote at x=-3, positive for x>3, negative for x<3
     
    V.
    No vertical asymptotes or holes, function passes through origin
    VI.
    Vertical asymptote at x=-3, positive for x<3, negative for x>3
  8. Determine, with support, an equation for a rational function of the form \( y = \dfrac{g(x)}{h(x)} \) that satisfies the given conditions.
    1. Vertical asymptotes of \( x = -1 \) and \( x = 3 \)
    2. Vertical asymptotes of \( x = -2 \) and \( x = \frac{-1}{2} \) and a \( y \)-intercept of \( \frac{3}{2} \)
    3. A hole at \( (\frac{1}{3},-2) \) and a vertical asymptote of \( x = 1 \)
    4. \( g(2) = 0 \) and \( h(2) = 0 \), a vertical asymptote of \( x = 2 \) and a hole at \( (\frac{1}{5},\frac{5}{6}) \)

  9. Determine an equation for the rational function shown in the graph below.

    Quadratic, opening down, vertex at (0,4), holes at (-1, 3), (2, 0)