12.4. Solved Problems

1. Find the asymptotes and axis intercepts of: f(x) = 2x/(x + 1)

            2x
   f(x) = -------
           x + 1
   

   There is one vertical asymptote at x = -1
   There is a horizontal asymptote at y = 2
   There is an x-axis intercept at x = 0
   The y-axis intercept is at y = f(0) = 0
   

2. Find the asymptotes and axis intercepts of: f(x) = 4/(4x + 3)

             4
   f(x) = ------
          4x + 3
   

   There is one vertical asymptote at x = -3/4
   There is a horizontal asymptote at y = 0
   There is no x-axis intercept
   The y-axis intercept is at y = f(0) = 4/3
   

3. Find the asymptotes and axis intercepts of: f(x) = -1/x²

           -1
   f(x) = ----
           x2
   

   There is one vertical asymptotes at x = 0
   There is a horizontal asymptote at y = 0
   There is no x-axis intercept
   There is no y-axis intercept
   

4. Find the asymptotes and axis intercepts of: f(x) = 6/(x² - 25)

                6
   f(x) = -------------
          (x + 5)(x - 5)
   

   There are two vertical asymptotes at x = -5, 5
   There is a horizontal asymptote at y = 0
   There is no x-axis intercept
   The y-axis intercept is at y = f(0) = -6/25
   

5. Find the asymptotes and axis intercepts of: f(x) = (x - 3)/(x² - 4)

               x - 3
   f(x) = --------------
          (x + 2)(x - 2)
   

   There are two vertical asymptotes at x = -2, 2
   There is a horizontal asymptote at y = 0
   The x-axis intercept is at x = 3
   The y-axis intercept is at y = f(0) = 0.75
   

6. Find the asymptotes and axis intercepts of: f(x) = (x + 1)/(2x² + 5x - 3)

               x + 1
   f(x) = --------------
          (x + 3)(2x - 1)
   

   There are two vertical asymptotes at x = -3, 0.5
   There is a horizontal asymptote at y = 0
   There is an x-axis intercept at x = -1
   The y-axis intercept is at y = f(0) = -1/3
   

7. Find the asymptotes and axis intercepts of: f(x) = (3x - 1)/(x² - 9)

               3x - 1
   f(x) = --------------
          (x + 3)(x - 3)
   

   There are two vertical asymptotes at x = -3, 3
   There is a horizontal asymptote at y = 0
   There is an x-axis intercept at x = 1/3
   The y-axis intercept is at y = f(0) = 1/9
   

8. Find the asymptotes and axis intercepts of: f(x) = (x - 5)/(3x² - 14x - 5)

               x - 5           1
   f(x) = --------------- = ------
          (3x + 1)(x - 5)   3x + 1
   

   There is one vertical asymptote at x = -1/3
   The function is undefined at x = 5
   There is a horizontal asymptote at y = 0
   There is no x-axis intercept
   The y-axis intercept is at y = f(0) = 1
   

9. Find the asymptotes and axis intercepts of: f(x) = x/(9 - x²)

                x 
   f(x) = --------------
          (3 + x)(3 - x)
   

   There are two vertical asymptotes at x = -3, 3
   There is a horizontal asymptote at y = 0
   There is an x-axis intercept at x = 0
   The y-axis intercept is at y = f(0) = 0
   

10. Graph: f(x) = 2x/(x² - 4)

                 2x 
    f(x) = --------------
           (x + 2)(x - 2)
    

    There are two vertical asymptotes at x = -2, 2
    There is a horizontal asymptote at y = 0
    There is an x-axis intercept at x = 0
    The y-axis intercept is at y = f(0) = 0
    

    

11. Find the asymptotes and axis intercepts of: f(x) = x²/(x² + 4x - 5)

                 x2
    f(x) = -------------
           (x + 5)(x - 1)
    

    There are two vertical asymptotes at x = -5, 1
    There is a horizontal asymptote at y = 1
    There is an x-axis intercept at x = 0
    The y-axis intercept is at y = f(0) = 0
    

12. Find the asymptotes and axis intercepts of: f(x) = (x² - 1)/(x² + 3x + 2)

           (x + 1)(x - 1)   (x - 1)
    f(x) = -------------- = -------
           (x + 2)(x + 1)   (x + 2)
    

    There is one vertical asymptote at x = -2
    The function is undefined at x = -1
    There is a horizontal asymptote at y = 1
    There is an x-axis intercept at x = 1
    The y-axis intercept is at y = f(0) = -1/2
    

13. Find the asymptotes and axis intercepts of: f(x) = (x² + 1)/(x² + x - 2)

               x2 + 1 
    f(x) = --------------
           (x + 2)(x - 1)
    

    There are two vertical asymptotes at x = -2, 1
    There is a horizontal asymptote at y = 1
    There is no x-axis intercept
    The y-axis intercept is at y = f(0) = -1/2
    

14. Find the asymptotes and axis intercepts of: f(x) = (x² - 4)/(x² + 3x - 10)

           (x + 2)(x - 2)    x + 2
    f(x) = -------------- = -------
           (x + 5)(x - 2)    x + 5
    

    There is one vertical asymptote at x = -5
    The function is undefined at x = 2
    There is a horizontal asymptote at y = 1
    There is an x-axis intercept at x = -2
    The y-axis intercept is at y = f(0) = 2/5
    

15. Find the asymptotes and axis intercepts of: f(x) = 3x²/(4x² - 1)

                 3x2 
    f(x) = ----------------
           (2x + 1)(2x - 1)
    

    There are two vertical asymptotes at x = -1/2, 1/2
    There is a horizontal asymptote at y = 3/4
    There is an x-axis intercept at x = 0
    The y-axis intercept is at y = f(0) = 0
    

16. Find the asymptotes and axis intercepts of: f(x) = (x² + 1)/(x - 1)

            x2 + 1              2
    f(x) = ------- = x + 1 + -------
            x - 1             x - 1
    

    There is one vertical asymptote at x = 1
    There is an oblique asymptote, y = x + 1
    There is no x-axis intercept
    The y-axis intercept is at y = f(0) = -1
    

17. Find the asymptotes and axis intercepts of: f(x) = (x³ - 27)/(x² - 9)

           (x - 3)(x2 + 3x + 9)    x2 + 3x + 9          9
    f(x) = -------------------- = ------------ = x + -----
              (x + 3)(x - 3)          x + 3          x + 3
    

    There is one vertical asymptote at x = -3
    The function is undefined at x = 3
    There is an oblique asymptote, y = x
    There is no x-axis intercept
    The y-axis intercept is at y = f(0) = 3
    

18. Find the asymptotes and axis intercepts of: f(x) = (x³ - 3x² - 4x)/(x² + 3x)

           x(x + 1)(x - 4)             14
    f(x) = --------------- = x - 6 + -------
              x(x + 3)                x + 3
    

    There is one vertical asymptote at x = -3
    The function is undefined at x = 0
    There is an oblique asymptote, y = x - 6
    There are x-axis intercepts at x = -1, x = 4
    

19. Graph: f(x) = x³/(x² - 1)

                 x3                x
    f(x) = -------------- = x + ------
           (x + 1)(x - 1)       x2 - 1
    

    There are two vertical asymptotes at x = -1, 1
    There is an oblique asymptote, y = x
    There is an x-axis intercept at x = 0
    The y-axis intercept is at y = f(0) = 0
    

    

20. Find the asymptotes and axis intercepts of: f(x) = (x² - x + 1)/(x - 1)

           x2 - x + 1         1
    f(x) = ---------- = x + -----
              x - 1         x - 1
    

    There is one vertical asymptote at x = 1
    There is an oblique asymptote, y = x
    There are no x-axis intercepts
    The y-axis intercept is at y = f(0) = -1
    

21. Graph: f(x) = (x³ - x² - x + 1)/(x² + 1)

           (x + 1)(x - 1)2            2x - 2
    f(x) = --------------- = x - 1 - -------
               x2 + 1                 x2 + 1
    

    There are no vertical asymptotes
    There is an oblique asymptote, y = x - 1
    There are x-axis intercepts at x = -1, 1
    The y-axis intercept is at y = f(0) = 1
    

    

22. Find the asymptotes and axis intercepts of: f(x) = 2x³/(x + 1)

             2x3
    f(x) = -------
            x + 1
    

    There is one vertical asymptote at x = -1
    There is no horizontal asymptote
    There is an x-axis intercept at x = 0
    The y-axis intercept is at y = f(0) = 0