Inverse Functions
November 30, 2012
Notes
Vocabulary
Inverse:

How to determine if two functions are inverses:
1. a. Show that the following functions are inverses using composition: f  x  
x 3
 and
4 2
3

g x  4 x   .
2

b. Now, graph the two functions on your calculator. Does the graph support your answer from
part a?
2. a. Show that the following functions are inverses using composition: f  x   2x  1 and
g x 
x 1
.
2
b. Now, graph the two functions on your calculator. Does the graph support your answer from
part a?
x
3. Graph the functions f  x   10 and g  x   log  x  . Are these functions inverses?
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Inverse Functions
November 30, 2012

How to find the inverse:

Notation:
4. Find the inverse of the following functions.
a. f  x   2x  5
b. g  x   10x 3
5. The following table shows the amount of garbage, G  t  , in millions of tons, produced in the
U.S. as reported by the EPA.
t: years since
1900
G t 
60
65
70
75
80
85
90
90
105
120
130
150
165
180
a. Evaluate G  90  .
b. Interpret your answer from part a within the context of the situation.
1
c. Evaluate G  90  .
d. Interpret your answer from part c within the context of the situation.
1
e. Evaluate G 120  .
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Inverse Functions
f.
November 30, 2012
Interpret your answer from part e within the context of the situation.
6. The height, h, of the ball thrown off of a building, in feet, as a function of time, t, in seconds is
modeled by the graph.
h
18
16
14
12
10
8
6
4
2
-2
t
1
2
3
4
a. Evaluate h  2 .
b. Interpret your answer from part a within the context of the situation.
c. Evaluate h1  7  .
d. Interpret your answer from part c within the context of the situation.
7. If P = f (t) gives the population of a city in thousands, as a function of time, t, in years since
2005,
a. What does f  5  represent?
1
b. What does f  5  represent?
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Inverse Functions
November 30, 2012
8. A college meal plan is based on a membership fee plus a price per meal. Thirty meals cost
$152.50 and 60 meals cost $250. Let P = f(n) where P is the price of the meal plan in dollars
and n is the number of meals.
a. Write an equation for P.
b. Complete the following statement:
 In the function f, _______________ is a function of _______________.
c. Find f 1  P  .
d. Complete the following statement:

In the function f 1 , _______________ is a function of _______________.
e. Evaluate f  80  and interpret what this means in the context of the situation.
f.
1
Evaluate f  80  and interpret what this means in the context of the situation.
4