# 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.

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.

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).

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.

Return to Exercises

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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