Answers to Questions on Rational Functions


    1. Question: Find the domain and any zeros, vertical asymptotes, horizontal asymptotes and holes for
      x - 4
      g(x) = --------------
      2x + 1

      Answer: Domain: All reals except -1/2, zero: x = 4, vertical asymptote: x = -1/2,
      horizontal asymptote: y = 1/2, no holes.
      Explain

    2. Question: Find the domain and any zeros, vertical asymptotes, horizontal asymptotes and holes for
      x
      f(x) = --------------
      x2 + 3

      Answer: Domain: All reals, zero: x = 0, no vertical asymptote,
      horizontal asymptote: y = 0, no holes. Explain

    3. Question: Find the domain and any zeros, vertical asymptotes, horizontal asymptotes and holes for
      x + 1
      r(x) = ------------------
      x2 - 3 x - 4

      Answer: Domain: all reals except x = 4, x = -1, vertical asymptote: x = 4,
      horizontal asymptote: y = 0, hole at (-1, -1/5). Explain

    4. Question: Find the domain and any zeros, vertical asymptotes, horizontal asymptotes and holes for
      x2 - 9
      h(x) = --------------
      x + 5

      Answer: Domain: all reals except x = -5, zeros: x = 3, x = -3,
      no horizontal asymptote, no holes. Explain

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  1. Scott and Ian design a cool T-shirt for snow boarders. Their friends are very impressed and everybody wants one, so Scott and Ian set up a T-shirt printing business in their garage. Total start-up costs are $450 due to the availability of a used graphics machine. They estimate that it costs them $5.50 to print each T-shirt.

    1. Question: Write a linear function C(x) giving the total cost of producing x T-shirts. Remember to take the start-up cost into account.

      Answer: C(x) = 450 + 5.5 x (total cost in dollars).

    2. Question: Write a rational function A(x) giving the average cost of producing x T-shirts.
      Answer:
      C(x) 450 + 5.5 x
      A(x) = -------- = ------------------
      x x

    3. Question: What is the domain of A(x) in the context of the problem? Explain.

      Answer: Domain is x > 0. Since x represents the number of shirts produced, only non-negative values make sense. We have to exclude x = 0 since A(x) is undefined for x = 0.

    4. Question: Does A(x) have a vertical asymptote? If so what is its equation?

      Answer: The vertical asymptote is x = 0.

    5. Question: Find the horizontal asymptote for A(x). What meaning does this value have in the context of the problem?

      Answer: The horizontal asymptote is y = 5.5. As the number of shirts produced increases, the start-up costs are "spread out" over a larger number of shirts. To see this mathematically, divide each term in the numerator by x: A(x) = 450/x + (5.5 x)/x = 450/x + 5.5. As x increases without bound, 450/x approaches zero (becomes negligible). Thus the average cost approaches, but never reaches $5.50 per T-shirt.

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