
Answers to Questions on Rational Functions
 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.
 Question: Find the domain and any zeros, vertical
asymptotes, horizontal asymptotes and holes for
 x 
f(x) =   
 x^{2} + 3 
Answer: Domain: All reals, zero: x = 0, no
vertical asymptote,
horizontal asymptote: y = 0, no holes.
 Question: Find the domain and any zeros, vertical
asymptotes, horizontal asymptotes and holes for
 x + 1 
r(x) =   
 x^{2}  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).
 Question: Find the domain and any zeros, vertical
asymptotes, horizontal asymptotes and holes for
 x^{2}  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
 Scott and Ian design a cool Tshirt for snow boarders.
Their friends are very impressed and everybody wants one, so Scott and
Ian set up a Tshirt printing business in their garage. Total startup
costs are $450 due to the availability of a used graphics machine. They
estimate that it costs them $5.50 to print each Tshirt.
 Question: Write a linear function
C(x) giving the total cost of producing x Tshirts. Remember to take
the startup cost into account.
Answer: C(x) = 450 + 5.5 x (total cost in
dollars).
 Question: Write a rational function A(x) giving
the average cost of producing x Tshirts.
Answer:  C(x)  450 + 5.5 x 
A(x) =   =   
 x  x 
 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 nonnegative values make sense. We have
to exclude x = 0 since A(x) is undefined for x = 0.
 Question: Does A(x) have a vertical asymptote? If
so what is its equation?
Answer: The vertical asymptote is x = 0.
 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 startup 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 Tshirt.
Return to Exercises
