Each function defined in Exercises 31–38 is a one-to-one function. Graph the function as a
solid line (or curve) and then graph its inverse on the same set of axes as a dashed line (or
curve). In Exercises 35–38 complete the table so that graphing the function will be easier. See
Example 4.
31.ƒ 1 x 2 = 2 x- 1 32.ƒ 1 x 2 = 2 x+ 3 33.g 1 x 2 =- 4 x 34.g 1 x 2 =- 2 x
SECTION 10.1 Inverse Functions 579
x
0
1
4ƒ 1 x 2 x
0
1
4ƒ 1 x 2- , 36. , 37.ƒ^1 x^2 =x^3 -^2 38.ƒ^1 x^2 =x^3 +^3
xÚ 0
ƒ 1 x 2 =- 2 x
xÚ 0ƒ 1 x 2 = 2 x
x0
1
2- 1
ƒ 1 x 2 x0
1- 1
- 2
ƒ 1 x 2EXERCISES 39 – 42FOR INDIVIDUAL OR GROUP WORK
Inverse functions can be used to send and receive coded
information. A simple example might use the function
defined by. ( Note that it is one-to-one.)
Suppose that each letter of the alphabet is assigned a
numerical value according to its position, as follows.Using the function, the word ALGEBRAwould be encoded as
7 2919159 417,
because
and so on.
The message would then be decoded by using the inverse of ƒ,defined by
For example,and so on.Work Exercises 39– 42 in order.
39.Suppose that you are an agent for a detective agency. Today’s function for your code
is defined by. Find the rule for algebraically.
40.You receive the following coded message today. (Read across from left to right.)Use the letter/number assignment described earlier to decode the message.
41.Why is a one-to-one function essential in this encoding/decoding process?
42.Use to encode your name, using the letter/number assignment
described earlier.ƒ 1 x 2 =x^3 + 443
31
- 1
- 1
51
11
43 75 55 67
75
3
7
75
67
79
27
15
23
75
7
7
15
15
- 1
71
11
27
67
71
31
15
43
51
- 1
31
67
27
51
- 1
15
7
59
23
15
47
31
23
95
71
71
23
75
31
ƒ 1 x 2 = 4 x- 5 ƒ-^1ƒ-^11292 =29 - 5
2
ƒ-^1172 = = 12 =L,7 - 5
2
= 1 =A,
Aor ƒ-^11 x^2 =
1
2 x-5
2 B.ƒ-^11 x 2 =x- 25ƒ 1 A 2 =ƒ 112 = 2112 + 5 =7, ƒ 1 L 2 =ƒ 1122 = 21122 + 5 =29,V
W
X
Y
Z
22
23
24
25
26
Q
R
S
T
U
17
18
19
20
21
L
M
N
O
P
12
13
14
15
16
G
H
I
J
K
7
8
9
10
11
A B C D E F
1 2 3 4 5 6
ƒ 1 x 2 = 2 x+ 5RELATING CONCEPTS
This is an Enigma
machine, used by the
Germans in World War II
to send coded messages.