Write Rational Functions - Problems With Solutions

Write rational functions given their characteristics such as vertical asymptotes, horizontal asymptote, x intercepts, hole.



Problem 1: Write a rational function f that has a vertical asymptote at x = 2, a horizontal asymptote y = 3 and a zero at x = - 5.

Solution to Problem 1:

  • Since f has a vertical is at x = 2, then the denominator of the rational function contains the term (x - 2). Function f has the form.

    f(x) = g(x) / (x - 2)

  • g(x) which is in the numerator must be of the same degree as the denominator since f has a horizontal asymptote. Also g(x) must contain the term (x + 5) since f has a zero at x = - 5. Hence

    f(x) = 3 (x + 5) / (x - 2)

  • Check that all the characteristics listed in the problem above are in the graph of f shown below.

    Graph of rational function, problem 1.

Problem 2: Write a rational function g with vertical asymptotes at x = 3 and x = -3, a horizontal asymptote at y = -4 and with no x intercept.

Solution to Problem 2:

  • Since g has a vertical is at x = 3 and x = -3, then the denominator of the rational function contains the product of (x - 3) and (x + 3). Function g has the form.

    g(x) = h(x) / [ (x - 3)(x + 3) ]

  • For the horizontal asymptote to exist, the numerator h(x) of g(x) has to be of the same degree as the denominator with a leading coefficient equal to -4. At the same time h(x) has no real zeros. Hence

    f(x) = [ -4x 2 - 6 ] / [ (x - 3)(x + 3) ]

  • Check the characteristics in the graph of g shown below.

    Graph of rational function, problem 2.

Problem 3: Write a rational function h with a hole at x = 5, a vertical asymptotes at x = -1, a horizontal asymptote at y = 2 and an x intercept at x = 2.

Solution to Problem 3:

  • Since h has a hole at x = 5, both the numerator and denominator have a zero at x = 5. Also the vertical asymptote at x = -1 means the denominator has a zero at x = -1. An x intercept at x = 2 means the numerator has a zero at x = 2. Finally the horizontal asymptote y = 2 means that the numerator and the denominator have equal degrees and the ratio of their leading coefficients is equal to 2. Hence

    h(x) = [ 2 (x - 5)(x - 2) ] / [ (x - 5)(x + 1) ]

  • The graph of h is shown below, check the characteristics.

    Graph of rational function, problem 3.

Problem 4: Write a rational function f with a slant asymptote y = x + 4, a vertical asymptote at x = 5 and one of the zeros at x = 2.

Solution to Problem 4:

  • The graph of f has a slant asymptote y = x + 4 and a vertical asymptote at x = 5, hence f(x) may be written as follows

    f(x) = (x + 4) + a / (x - 5)

  • where a is a constant to be determined using the fact that f(2) = 0 since f has a zero at x = 2.

    f(2) = (2 + 4) + a / (2 - 5) = 0

  • Solve the above for a to obtain.

    a = 18

  • Hence f(x) is given by.

    f(x) = (x + 4) + 18 / (x - 5) = (x 2 - x - 2) / (x - 5)

  • Check the characteristics of the graph of f shown below.

    Graph of rational function, problem 4.

More math problems with detailed solutions in this site.


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Updated: 2 April 2013

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