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Module 1
Domain and Range from Graphs
18
For the graphs, state the domain and range in interval notation.
1.
2.





































3.
4.











































5.
6.










































Module 1
Analyzing the Graph of a Function
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1. For the graph f(x) below, find the indicated information.
(6,18)
18
16
14
12
10
8
(-2,4)
6
4
2
(-4.0)
-6
-4
(4.0)
(0,0)
-2
2
-2
-4
-6
-8
(2,-6)
-10
(-6,-12)
a. Find f(0).
-12
-14
b. Find f(2).
c. Find f(–6).
3
d. Is f   positive or negative?
2
e. Is f(–1) positive or negative?
f. For what numbers is f(x) = 0?
g. For what numbers is f(x) < 0?
h. What is the domain of f(x)?
i. What is the range of f(x)?
j. What are the coordinates of the x-intercepts?
k. What are the coordinates of the y-intercept?
l. How often does the line y = –3 intersect the graph?
m. How often does the line x = 1 intersect the graph?
n. Find a value of x for which f(x) = 7.
4
6
Module 1
Analyzing the Graph of a Function
20
2. For the graph g(x) below, find the indicated information. Write answers in interval
notation where appropriate.




      











a. Find g(5).
b. State the intervals of x where the graph is increasing.
c. State the intervals of x where the graph is decreasing.
d. State the interval of x where the graph is constant.
e. State the coordinates of the x-intercept(s).
f. State the coordinates of the y-intercept.
g. State the domain.
h. State the range.
i. Find any local maxima.
j. Find any local minima.

Module 1
Linear Functions
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1. Suppose that 650 lb of beans are sold at a price of $4 per pound, and 400 lb
are sold at $8 per pound.
a. List the data points (use price as the independent variable).
b. Find the rate of change (slope of the line joining the points).
c. Use the point-slope form to write a linear equation for this data.
Write the answer in slope-intercept form.
d. Use this equation to predict how many pounds of beans consumers would be willing
to buy at $5.60 per pound.
2. A man who is moving into a new apartment plans to rent a truck from a rental
company. The cost to rent the truck is $39.95 for the entire day plus $0.35 per mile.
a. Write a linear equation that expresses the cost of renting the truck based on the
number of miles he drives. (Use number of miles as the independent variable.)
b. If the distance from the man’s old apartment to his new apartment is 12 miles,
what will be the cost to drive the truck from his old apartment to his new one and
back again?
c. If the man has budgeted $90 for rental expenses, how many miles can he drive
the truck and keep his costs within his budget?
3. A woman needs to hire a plumber and finds that two co-workers have used someone
that they consider reliable. The first co-worker used this plumber for a 2-hour job at a
cost of $140. The other co-worker used this plumber for a 5-hour job at a cost of
$275. Assume the cost of hiring the plumber follows a linear equation.
a. Use the point-slope form to find an equation that gives the cost of the job as a
function of the time spent on the job.
b. Predict the cost of a 4-hour job.
Module 1
Transformations of Functions
1. Graph each of the following functions.
Group 1: y1 = x3
Group 2: y1 = │x│
3
y2 = x + 2
y2 = │x│+ 2
3
y3 = x – 1
y3 = │x│– 1
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Group 3: y1 =
y2 =
y3 =
x
x+2
x–1
a. Compare the graphs of y1 = f(x) and y2 = f(x) + c where c represents a real number.
b. Describe the change in the graph when a positive number is added.
c. Describe the change in the graph when a negative number is added.
2. Graph each of the following functions.
Group 1: y1 = x3
Group 2: y1 = │x│
y2 = (x + 2)3
y2 = │x + 2│
3
y3 = (x – 1)
y3 = │x – 1│
Group 3: y1 =
y2 =
y3 =
x
x2
x 1
a. Compare the graphs of y1 = f(x) and y2 = f(x + c) where c represents a real number.
b. Describe the change in the graph when c is a positive number.
c. Describe the change in the graph when c is a negative number.
3. Graph each of the following functions.
Group 1: y1 = x3
Group 2: y1 = │x│
3
y2 = –x
y2 = –│x│
y3 = (–x)3
y3 = │–x │
Group 3: y1 = x
y2 = – x
y3 =  x
a. Compare the graphs of y1 = f(x) and y2 = –f(x). Describe the differences in the graphs.
b. Compare the graphs of y1 = f(x) and y3 = f(–x). Describe the differences in the graphs.
4. Graph each of the following functions.
Group 1: y1 = x3
Group 2: y1 = │x│
3
y2 = 2x
y2 = 2│x │
y3 = 0.5x3
y3 = 0.5│x│
Group 3: y1 = x
y2 = 2 x
y3 = 0.5 x
a. Compare the graphs of y1 = f(x) and y2 = c f(x).
b. Describe the change in the graphs where c is a positive number greater than 1(c > 1).
c. Describe the change in the graphs where c is a positive number less than 1 (c < 1).
5. Graph each of the following functions.
Group 1: y1 = x3
Group 2: y1 = │x│
Group 3: y1 = x
3
y2 = (2x)
y2 = │2x │
y2 = 2x
y3 = (0.5x)3
y3 = │0.5x│
y3 = 0.5x
a. Compare the graphs of y1 = f(x) and y2 = f(cx).
b. Describe the change in the graphs where c is a positive number greater than 1 (c > 1).
c. Describe the change in the graphs where c is a positive number less than 1 (c < 1).
Module 1
Transformations of Functions continued
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6. Using the information about the basic graphs of y = x3, y = x and y = │x│,
sketch a graph of the following functions. Use a calculator to check the graphs.
a. ya = (x + 2)3 – 1
b. yb = │x + 3│ + 2
c. yc = x  1  3
7. Write the equations of each of the following graphs. Use a calculator to check
the graphs.
8
8
5
7
7
4
6
6
5
5
4
4
3
3
2
2
3
2
1
-7
-6
-5
-4
-3
-2
-1
1
1
2
3
4
5
6
7
8
9
10
-8
-7
-6
-5
-4
-3
-2
-1
-1
1
1
-1
2
3
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7
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9
-8
-7
-6
-5
-4
-3
-2
-1
-1
-2
-2
-2
-3
-3
-3
-4
-4
-5
-5
-6
-6
-7
-7
-8
-8
-9
-9
-4
-5
-6
1
2
3
4
5
6
7
8. The graph of a function p(x) is shown. Draw the graph of –p(x–2) + 3.
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7
6
5
4
3
2
1
-8
-7
-6
-5
-4
-3
-2
-1
-1
1
2
3
4
5
6
7
8
9
-2
-3
-4
-5
-6
-7
-8
-9
9. Sketch a graph of the function a(x) = 2 x .
10. Write the equation of the function that has a graph the shape of y = 3 x , vertically
shrunk by a factor of ¼ and shifted right 6 units. Then, sketch the graph.
11. Write the equation of the function that has a graph the shape of y = │x│, reflected
about the x-axis and shifted down 1 unit. Then, sketch the graph.
12. Describe the transformations that have been applied to the graph of F(x) = x2 to obtain
the graph of f(x)  3(x)2  5 .
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9
Module 1
Combinations of Functions
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1. For f(x)  3x 2  5 and g(x)  7x  2 ,
a. Verify: g(x + 2) ≠ g(x) + g(2).
b. Find (f – g)(x).
c. Using the resulting function in (b), show that (f – g)(2) = f(2) – g(2).
(The work should be different for each side of the equation.)
f
d. Is (fg)(0) =   (0)? Explain.
 g
e. Find
f(x  h)  f(x)
, h ≠ 0.
h
2. For the graph, answer the following questions.
y


g(x)



f(x)


x






a. Determine (f + g)(1).
b. Determine (f – g)(6).
c. Determine (f g)(2).
f
d. Determine   (3).
 g
e. State the domain of f(x).
f. State the domain of g(x).
g. State the domain of (f + g)(x).





Module 1
Composite Functions
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1. A size 36 shoe in France is size 3.5 in England. A function that converts shoe sizes
3x  94
in France to those in England is g(x) 
. A size 6 shoe in the United States
4
is size 3.5 in England. A function that converts shoe sizes in England to those in the
5
United States is f(x) = x + .
2
a. Use composition of functions to find a function that converts shoe sizes in France to
those in the United States. Simplify the function.
b. Determine the shoe size in the United States for a size 38 shoe in France.
2. A stone is dropped into a liquid forming circles which increase in radius with time
according to the formula r(t) = 4t. The area of a circle related to radius is A(r)   r 2 .
Using composition of functions, find an equation that would give area as a function of
time.
3. To convert from x degrees Celsius to y degrees Fahrenheit, use the formula
9
y = f(x) = x  32 .
5
a. Find the inverse function g(x) that converts from x degrees Fahrenheit to y degrees
Celsius.
b. Find (f g)( x) and (g f )( x ) to verify that f(x) and g(x) are inverse functions.
4. As the number of units sold increases, market price decreases (supply and demand).
Suppose that p = 5000 – 0.75x , where p is the market price and x is the number of
units sold. Suppose further that the cost of producing x items is given by
C(x) = 3000 + 15x, and that the revenue from the sale of x units is given by
R(x) = 120x.
a. Express the cost as a function of price, p.
b. Express revenue as a function of price, p.
c. Given that profit is revenue minus cost, express profit as a function of price, p.
Module 1
Inverse Functions
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Cryptology is the science of making and breaking codes. This exercise explores how the
idea of functions and their inverses can be used to encode and decode messages.
To encode and decode a message, first replace each letter of the alphabet with a positive
integer using the following scheme, thus rewriting the original message as numbers instead
of words:
A–1
B–2
C–3
D–4
E–5
F–6
G–7
H–8
I–9
J – 10
K – 11
L – 12
M – 13
N – 14
O – 15
P – 16
Q – 17
R – 18
S – 19
T – 20
U – 21
V – 22
W – 23
X – 24
Y – 25
Z – 26
Blank - 27
ENCODING: A one-to-one function can be used to encode a numerical message.
For example, suppose you want to send the message MATH to a friend, and you have
decided that the function f(x) = 3x + 4 will be the encoding function. This function simply
describes the procedure used to create the encoded message – in this case multiply by
3 and add 4. First change the letters to corresponding numbers as shown above:
13 1 20 8. Then use these as the input values in f(x).
f(13) = 3(13) + 4 = 43
f(1) = 7
f(20) = 64
f(8) = 28
So the encoded message that you send to your friend is: 43 7 64 28
NOTE: A graphing calculator can be used to evaluate a function as above:
a. Press [Y=] on the calculator. Type 3x + 4.
b. Choose [2nd][WINDOW] (TBLSET). Set the independent variable to Indpnt: Ask.
c. Then choose [2nd][GRAPH] (TABLE). Input each x value (13 1 20 8). The calculator
will return the function values (43 7 64 28) which is the encoded message.
Module 1
Inverse Functions (continued)
27
DECODING an encoded message: Now it is up to your friend to decode the message.
Decoding is the process that “undoes” the encoding process. If f(x) encodes the message,
what will decode it? The inverse of f(x) or f -1(x)! So f-1(x) will be the decoding function.
x4
. Take the
3
encoded message (43 7 64 28) and use these values as input values in f -1(x). Again, the
x4
calculator can be used to decode – simply enter y =
and use the TABLE feature as
3
described above.
In the example, the inverse of f(x) = 3x + 4 can be shown to be f-1(x) =
f-1(43) = 13
f-1(7) = 1
f-1(64) = 20
f-1(28) = 8
That’s the original numerical message! The last step is to convert back to letters using the
table given previously. Now your friend knows the message that you sent: MATH.
Cryptology Applications:
1. Assume that the encoding function is f(x) = 6x – 2.
a. Find the inverse of f(x).
b. Using the inverse function you just found, decode the following encoded
message. The answer should be words. Put spaces where appropriate!
16 88 70 70 28 40 28 160 4 70 40 28 10 106 4 160 52 112 160 76
148 160 34 4 130 88 106 52 118 28 160 112 124 10 58 28 16 118
2. Suppose f(x) = 2x + 7 is the encoding function.
a. Encode “Collin Cougars”. The answer will be numbers.
b. Find the inverse of f(x).
c. Decode the following. The answer will be words.
47 23 17 61 47 25 33 17 61 25 45 61 35 37 53
3. Why is it necessary for the encoding function to be one-to-one?