Homework 3
Solutions to bold face problems
Section 11.2:
28 Find
functions f ◦ g and g ◦ f , where f (x) =
√ the rules for the composite
2
2 x + 3 and g(x) = x − 1.
Answer: We begin with f ◦ g. In this case
f ◦ g(x) = f (g(x))
q
= 2 g(x) + 3
√
= 2 x2 − 1 + 3.
For g ◦ f , we have
g ◦ f (x) =
=
=
=
=
g(f (x))
(f (x))2 − 1
√
(2 x + 3)2 − 1 Leaving the answer like this is fine
√
4x + 12 x + 9 − 1
√
4x + 12 x + 8
Please note that you do not need to simplify the answer for g ◦ f , and
you are probably just as well off to leave it as in the 3rd line above.
√
32 Evaluate h(2) where h = g ◦f , and f (x) = 3 x2 − 1 and g(x) = 3x3 +1.
Answer: The easy way to do this is to simply substitute in values so
that
√
3
f (2) =
22 − 1
√
3
=
3.
We now substitute this value into
√
3
g( 3) =
=
=
the equation for g(x) to get
√
3
3( 3)3 + 1
3(3) + 1
10.
Thus h(2) = g ◦ f (2) = g(f (2)) = 10.
1
A second method is to use the equations for f (x) and g(x) to find the
equation for h(x). This method looks as follows:
h(x) =
=
=
=
=
=
g ◦ f (x)
g(f (x))
3(f (x))3 + 1
√
3
3( x2 − 1)3 + 1
3(x2 − 1) + 1
3x2 − 2.
Now we can substitute x = 2 into this equation to get
h(2) = 3(22 ) − 2 = 10.
38 Find functions f and g such that h = g ◦ f where h(x) = (2x − 3)3/2 .
Answer: The method to do this often involves looking inside the
parentheses (or root sign), so in this case, we identify the “inner” function as f (x) = 2x − 3 and then h(x) = (f (x))3/2 . Then the “outer”
function would be g(x) = x3/2 . Thus an answer could be f (x) = 2x − 3
and g(x) = x3/2 .
56 The total cost incurred by time t in a production of a certain commodity is f (t) dollars. The number of products produced by time t is g(t)
units. What does the function f (t)/g(t) represent.
Answer: Following the descriptors (dollars and units), f (t)/g(t) would
have descriptor dollars/unit. Thus f (t)/g(t) denotes a type of cost/unit.
In this case, f (t)/g(t) denotes the total cost incurred by time t divided
by the number of units produced by time t. We would more commonly
call this the average cost of producing a unit between time 0
and time t.
Section 11.3
8 An efficiency study showed that the average worker at Delphi Electronics assembled cordless telephones at the rate of
3
f (t) = − t2 + 6t + 10 (0 ≤ t ≤ 4)
2
2
phones/hour, t hour after starting work during the morning shift. At
what rate does the average worker assemble telephones 2 hours after
starting work?
Answer: This is given by the value of f (2), which is
3
f (2) = − 22 + 6 · 2 + 10 = −6 + 12 + 10 = 16
2
phones/hour.
26 According to the World Wildlife Fund, a group in the forefront of the
fight against illegal ivory trade, the price of ivory (in dollars/kilo) compiled from a variety of legal and black market sources is approximated
by the function
(
f (t) =
8.37t + 7.44 if 0 ≤ t ≤ 8
2.84t + 51.68 if 8 < t ≤ 30
where t is measured in years, with t = 0 corresponding to the beginning
of 1970.
a Sketch the graph of the function f .
120
100
80
60
40
20
5
10
15
20
25
30
b What was the price of ivory at the beginning of 1970? At the
beginning of 1990?
Answer: In 1970, t = 0 and in 1990 t = 20. Thus in 1970, the
price is given by
f (0) = 8.37 · 0 + 7.44 = 7.44dollars/kilo,
3
and in 1990 the price is given by
f (20) = 2.84(20) + 51.68 = 108.48dollars/kilo
4