Coburn J.W. Algebra and Trigonometry
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270 CHAPTER 2 Relations, Functions, and Graphs 2-120
92. Given and graph
each function on the same axes by plotting the
points that correspond to integer inputs for
Do you notice anything? Next, find
and What
happened? Look closely at the functions f and g to
see how they are related. Can you come up with two
additional functions where the same thing occurs?
93. Given and , what
can you say about the domain of ? Enter
the functions as Y
1
and Y
2
on a graphing calculator,
then enter See if you can determine
why the calculator gives an error message for Y
3
,
regardless of the input.
94. Given , and
, (a) find the new function rule for
h and (b) determine the implied domain of h. Does
this implied domain include , and
as valid inputs? (c) Determine the actual
domain for and discuss the result.
95. Instead of calculating the result of an operation on
two functions at a specific point as in Exercises
69–74, we can actually graph the function that
h1x2 1f g21x2
x 3
x 2, x 2
h1x2 1f g21x2
f 1x2
1
x
2
4
, g1x2 1x 1
Y
3
Y
1
Y
2
.
1f g21x2
g1x2 1x 2
f 1x2 11 x
H1x2 1g f 21x2.h1x2 1f g21x2
x 33, 34.
g1x2 2
3
x 2,f 1x2 x
3
2
results from the operation. This skill, called the
addition of ordinates, is widely applied in a study
of tides and other areas. For
and , complete a table of values
like the one shown for . For the last
column, remember that ,
and use this relation to complete the column.
Finally, use the ordered pairs to
graph the new function. Is the new function
smooth? Is the new function continuous?
1x, 1f g21x22
1f g21x2 f 1x2 g1x2
x 32, 84
g1x2 4
x 3
5
f 1x2 1x 32
2
2
xf(x) g(x)(f g)(x)
2
1
0
1
2
3
4
5
6
7
8
Exercise 95
MAINTAINING YOUR SKILLS
96. (1.4) Find the sum and product of the complex
numbers and
97. (2.4) Draw a sketch of the functions
(a) (b) , and
(c) from memory.h1x2
x
g1x2 2
3
xf 1x2 1x,
2 3i.2 3i
98. (1.5) Use the quadratic formula to solve
99. (2.3) Find the equation of the line
perpendicular to , that
also goes through the origin.
2x 3y 9
2x
2
3x 4 0.
SUMMARY AND CONCEPT REVIEW
SECTION 2.1 Rectangular Coordinates; Graphing Circles and Other Relations
KEY CONCEPTS
•
A relation is a collection of ordered pairs (x, y) and can be given in set or equation form.
•
As a set of ordered pairs, the domain of the relation is the set of all first coordinates, and the range is the set of
all corresponding second coordinates.
•
A relation can be expressed in mapping notation , indicating an element from the domain is mapped to
(corresponds to or is associated with) an element from the range.
•
The graph of a relation in equation form is the set of all ordered pairs (x, y) that satisfy the equation. We plot a
sufficient number of points and connect them with a straight line or smooth curve, depending on the pattern formed.
•
The midpoint of a line segment with endpoints (x
1
, y
1
) and (x
2
, y
2
) is .
•
The distance between the points (x
1
, y
1
) and (x
2
, y
2
) is .
•
The equation of a circle centered at (h, k) with radius r is .1x h2
2
1y k2
2
r
2
d 21x
2
x
1
2
2
1y
2
y
1
2
2
a
x
1
x
2
2
,
y
1
y
2
2
b
x S y
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EXERCISES
1. Represent the relation in mapping notation, then state the domain and range.
2. Graph the relation by completing the table, then state the domain and range of the relation.
Mr. Northeast and Mr. Southwest live in Coordinate County and are good friends. Mr. Northeast lives at 19 East
25 North or (19, 25), while Mr. Southwest lives at 14 West and 31 South or . If the streets in Coordinate
County are laid out in one mile squares,
3. Use the distance formula to find how far apart they live.
4. If they agree to meet halfway between their homes, what are the coordinates of their meeting place?
5. Sketch the graph of .
6. Sketch the graph of . Clearly state the center and radius.
7. Find the equation of the circle whose diameter has the endpoints and (0, 4).
SECTION 2.2 Graphs of Linear Equations
KEY CONCEPTS
•
A linear equation can be written in the form , where a and b are not simultaneously equal to 0.
•
The slope of the line through (x
1
, y
1
) and (x
2
, y
2
) is , where .
•
Other designations for slope are .
•
Lines with positive slope ( ) rise from left to right; lines with negative slope ( ) fall from left
to right.
•
The equation of a horizontal line is ; the slope is .
•
The equation of a vertical line is ; the slope is undefined.
•
Lines can be graphed using the intercept method. First determine (x, 0) (substitute 0 for y and solve for x), then
(0, y) (substitute 0 for x and solve for y). Then draw a straight line through these points.
•
Parallel lines have equal slopes ( ); perpendicular lines have slopes that are negative reciprocals
( or ).
EXERCISES
8. Plot the points and determine the slope, then use the ratio to find an additional point on the line:
a. and and b. (3, 4) and
9. Use the slope formula to determine if lines L
1
and L
2
are parallel, perpendicular, or neither:
a. L
1
: and (0, 6); L
2
: (1, 8) and (0, 5)
b. L
1
: (1, 10) and : L
2
: and
10. Graph each equation by plotting points: (a) and (b)
11. Find the intercepts for each line and sketch the graph: (a) and (b) y
4
3
x 2.2x 3y 6
y
3
2
x 1.y 3x 2
11, 3212, 1211, 72
12, 02
16, 12.15, 2214, 32
¢y
¢x
rise
run
m
1
#
m
2
1m
1
1
m
2
m
1
m
2
x h
m 0y k
m 6 0m 7 0
m
rise
run
change in y
change in x
¢y
¢x
vertical change
horizontal change
x
1
x
2
m
y
2
y
1
x
2
x
1
ax by c
13, 02
x
2
y
2
6x 4y 9 0
x
2
y
2
16
114, 312
y 225 x
2
517, 32, 14, 22, 15, 12, 17, 02, 13, 22, 10, 826
xy
0
2
4
5
2
4
5
2-121 Summary and Concept Review 271
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12. Identify each line as either horizontal, vertical, or neither, and graph each line.
a. b. c.
13. Determine if the triangle with the vertices given is a right triangle: (7, 2), (0, 16).
14. Find the slope and y-intercept of the line shown and discuss the slope ratio in this context.
SECTION 2.3 Linear Graphs and Rates of Change
KEY CONCEPTS
•
The equation of a nonvertical line in slope-intercept form is or . The slope of the line
is m and the y-intercept is (0, b).
•
To graph a line given its equation in slope-intercept form, plot the y-intercept, then use the slope ratio to
find a second point, and draw a line through these points.
•
If the slope m and a point (x
1
, y
1
) on the line are known, the equation of the line can be written in point-slope
form: .
•
A secant line is the straight line drawn through two points on a nonlinear graph.
•
The notation literally means the quantity measured along the y-axis is changing with respect to changes
in the quantity measured along the x-axis.
•
The average rate of change on the interval containing x
1
and x
2
is the slope of the secant line through (x
1
, y
1
) and
(x
2
, y
2
), or .
EXERCISES
15. Write each equation in slope-intercept form, then identify the slope and y-intercept.
a. b.
16
. Graph each equation using the slope and y-intercept.
a. b.
17. Graph the line with the given slope through the given point.
a. b.
18. What are the equations of the horizontal line and the vertical line passing through Which line is the
point (7, 5) on?
19. Find the equation of the line passing through (1, 2) and Write your final answer in slope-intercept form.
20. Find the equation for the line that is parallel to and passes through the point (3, 4). Write your final
answer in slope-intercept form.
21. Determine the slope and y-intercept of the line shown. Then write the equation of the line in
slope-intercept form and interpret the slope ratio in the context of this exercise. m
¢W
¢R
4x 3y 12
13, 52.
12, 52?
m
1
2
; 12, 32m
2
3
; 11, 42
h1x2
5
2
x 3f 1x2
2
3
x 1
5x 3y 154x 3y 12 0
¢y
¢x
y
2
y
1
x
2
x
1
m
¢y
¢x
y y
1
m1x x
1
2
m
¢y
¢x
f 1x2 mx by mx b
8246
2
4
6
8
10
Hawk population (100s)
Rodent population
(1000s)
x
y
15, 42,
2y x 5y 4x 5
10862
2
4
6
8
10
4
Wolf population (100s)
Rabbit population (100s)
R
W
272 CHAPTER 2 Relations, Functions, and Graphs 2-122
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22. For the graph given, (a) find the equation of the line in point-slope form, (b) use the
equation to predict the x- and y-intercepts, (c) write the equation in slope-intercept form,
and (d) find y when , and the value of x for which
SECTION 2.4 Functions, Function Notation, and the Graph of a Function
KEY CONCEPTS
•
A function is a relation, rule, or equation that pairs each element from the domain with exactly one element of
the range.
•
The vertical line test says that if every vertical line crosses the graph of a relation in at most one point, the relation
is a function.
•
On a graph, vertical boundary lines can be used to identify the domain, or the set of “allowable inputs” for a
function.
•
On a graph, horizontal boundary lines can be used to identify the range, or the set of y-values (outputs) generated
by the function.
•
When a function is stated as an equation, the implied domain is the set of x-values that yield real number outputs.
•
x-values that cause a denominator of zero or that cause the radicand of a square root expression to be negative
must be excluded from the domain.
•
The phrase “y is a function of x,” is written as . This notation enables us to evaluate functions while
tracking corresponding x- and y-values.
EXERCISES
23. State the implied domain of each function:
a. b.
24. Determine , and h(3a) for
25. Determine if the mapping given represents a function. If not, explain how the definition of a function is violated.
26. For the graph of each function shown, (a) state the domain and range, (b) find the value of f(2), and (c) determine
the value(s) of x for which .
I. II. III.
55
5
5
x
y
55
5
5
x
y
55
5
5
x
y
f 1x2 1
Apollo
Jupiter
Ares
Neptune
Mercury
Venus
Ceres
Mars
messenger
war
craftsman
love and beauty
music and healing
oceans
all things
agriculture
Mythological
deities
Primary
concern
h1x2 2x
2
3x.h122, h1
2
3
2
g1x2
x 4
x
2
x 6
f 1x2 24x 5
y f 1x2
y 15.x 20
2-123 Summary and Concept Review 273
108620
20
40
60
80
100
4
x
y
College Algebra—
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274 CHAPTER 2 Relations, Functions, and Graphs 2-124
SECTION 2.5 Analyzing the Graph of a Function
KEY CONCEPTS
•
A function f is even (symmetric to the y-axis), if and only if when a point (x, y) is on the graph, then ( ) is
also on the graph. In function notation: .
•
A function f is odd (symmetric to the origin), if and only if when a point (x, y) is on the graph, then ( ) is
also on the graph. In function notation: .
Intuitive descriptions of the characteristics of a graph are given here. The formal definitions can be found within
Section 2.5.
•
A function is increasing in an interval if the graph rises from left to right (larger inputs produce larger outputs).
•
A function is decreasing in an interval if the graph falls from left to right (larger inputs produce smaller outputs).
•
A function is positive in an interval if the graph is above the x-axis in that interval.
•
A function is negative in an interval if the graph is below the x-axis in that interval.
•
A function is constant in an interval if the graph is parallel to the x-axis in that interval.
•
A maximum value can be a local maximum, or global maximum. An endpoint maximum can occur at the
endpoints of the domain. Similar statements can be made for minimum values.
•
For any function f, the average rate of change in the interval [x
1
, x
2
] is .
•
The difference quotient for a function f(x) is .
EXERCISES
State the domain and range for each function f(x) given. Then state the intervals where f is increasing or decreasing
and intervals where f is positive or negative. Assume all endpoints have integer values.
27. 28. 29.
30. Determine which of the following are even , odd , or neither.
a. b.
c. d.
31a. Given , find the average rate of change in the interval . What does the result confirm
about the graph of this toolbox function?
31b. Use the difference quotient to find a rate of change formula for the function given, then calculate the rate of
change for the interval indicated: ; [2.00,2.01].
32. Draw the function f that has all of the following characteristics, then name the zeroes of the function and the
location of all maximum and minimum values. [Hint: Write them in the form (c, f(c)).]
a. Domain: b. Range:
c. d.
e. f. for
g. for
x 10, 62 ´ 19, 102f 1x27 0
x 16, 02 ´ 16, 92f 1x26 0f 1x2c for x 13, 32 ´ 17.5, 102
f 1x2T for x 16, 32 ´ 13, 7.52f 102 0
y 18, 62x 36, 102
j1x2 x
2
x
33, 54f 1x2 2x 4
q1x2
x
2
x
x
p1x2
3x
x
3
g1x2 x
4
2
3
x
x
f 1x2 2x
5
2
3
x
3f 1k2f 1k243f 1k2 f 1k24
1010
10
10
x
y
55
5
5
x
f(x)
y
1010
10
10
x
y
f 1x h2 f 1x2
h
f 1x
2
2 f 1x
1
2
x
2
x
1
f 1x2f1x2
x, y
f 1x2 f 1x2
x, y
College Algebra—
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SECTION 2.6 The Toolbox Functions and Transformations
KEY CONCEPTS
•
The toolbox functions and graphs commonly used in mathematics are
•
the identity function
•
squaring function: , parabola
•
square root function:
•
absolute value function:
•
cubing function:
•
cube root function:
•
For a basic or parent function , the general equation of the transformed function is .
For any function and h, ,
•
the graph of is the graph
•
the graph of is the graph of
of shifted upward k units shifted downward k units
•
the graph of is the graph of
•
the graph of is the graph of
shifted left h units shifted right h units
•
the graph of is the graph of
•
the graph of is the graph of
reflected across the x-axis reflected across the y-axis
•
results in a vertical stretch
•
results in a vertical compression
when when
•
Transformations are applied in the following order: (1) horizontal shifts, (2) reflections,
(3) stretches or compressions, and (4) vertical shifts.
EXERCISES
Identify the function family for each graph given, then (a) describe the end behavior; (b) name the x- and y-intercepts;
(c) identify the vertex, initial point, or point of inflection (as applicable); and (d) state the domain and range.
33. 34. 35.
36. 37.
Identify each function as belonging to the linear, quadratic, square root, cubic, cube root, or absolute value family.
Then sketch the graph using shifts of a parent function and a few characteristic points.
38. 39. 40.
41. 42.
43. Apply the transformations indicated for the graph of f(x) given.
a.
b.
c.
1
2
f 1x2
f 1x2 4
f 1x 22
f 1x2 2
3
x 2f 1x2 1x 5 2
f 1x2 x
3
1f 1x2 2
x 3
f 1x21x 22
2
5
55
5
x
y
5
55
5
x
y
5
55
5
x
y
5
55
5
x
y
5
55
5
x
y
5
0 6 a 6 1a 7 1
y af 1x2y af 1x2
y f 1x2y f 1x2
y f 1x2y f 1x2
y f 1x2y f 1x2
y f 1x h2y f 1x h2
y f 1x2y f 1x2
y f 1x2 ky f 1x2 k
k 7 0y f 1x2
y af 1x h2 ky f 1x2
f 1x2 2
3
xf 1x2 x
3
f 1x2
x
f 1x2 1x
f 1x2 x
2
f 1x2 x
2-125 Summary and Concept Review 275
55
5
5
y
f(x)
(4, 3)
(1.5, 1)
(1, 3)
x
College Algebra—
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SECTION 2.7 Piecewise-Defined Functions
KEY CONCEPTS
•
Each piece of a piecewise-defined function has a domain over which that piece is defined.
•
To evaluate a piecewise-defined function, identify the domain interval containg the input value, then use the piece
of the function corresponding to this interval.
•
To graph a piecewise-defined function you can plot points, or graph each piece in its entirety, then erase portions
of the graph outside the domain indicated for each piece.
•
If the graph of a function can be drawn without lifting your pencil from the paper, the function is continuous.
•
A pointwise discontinuity is said to be removable because we can redefine the function to “fill the hole.”
•
Step functions are discontinuous and formed by a series of horizontal steps.
•
The floor function gives the first integer less than or equal to x.
•
The ceiling function is the first integer greater than or equal to x.
EXERCISES
44. For the graph and functions given, (a) use the correct notation to write the relation as a
single piecewise-defined function, stating the effective domain for each piece by inspecting
the graph; and (b) state the range of the function:
45. Use a table of values as needed to graph h(x), then state its domain and range. If the function
has a pointwise discontinuity, state how the second piece could be redefined so that a
continuous function results.
46. Evaluate the piecewise-defined function p(x): , and
47. Sketch the graph of the function and state its domain and range. Use transformations of the toolbox functions
where possible.
48. Many home improvement outlets now rent flatbed trucks in support of customers that purchase large items. The
cost is $20 per hour for the first 2 hr, $30 for the next 2 hr, then $40 for each hour afterward. Write this information
as a piecewise-defined function, then sketch its graph. What is the total cost to rent this truck for 5 hr?
SECTION 2.8 The Algebra and Composition of Functions
KEY CONCEPTS
•
The notation used to represent the basic operations on two functions is
••
••
•
The result of these operations is a new function h(x). The domain of h is the intersection of domains for f and g,
excluding values that make for .h1x2 a
f
g
b1x2g1x2 0
a
f
g
b1x2
f 1x2
g1x2
; g1x2 01f
#
g21x2 f 1x2
#
g1x2
1f g21x2 f 1x2 g1x21f g21x2 f 1x2 g1x2
q1x2 •
21x 3 4 x 3
2
x
2 3 6 x 6 3
21x 3 4 x 3
p1x2 •
4 x 6 2
x
2 2 x 6 3
31x
9 x 3
p13.52p142, p122, p12.52, p12.992, p132
h1x2 •
x
2
2x 15
x 3
x 3
6
x 3
Y
3
31x 3 1.
Y
1
5, Y
2
x 1,
<x=
:x;
276 CHAPTER 2 Relations, Functions, and Graphs 2-126
1010
10
10
y
x
Y
1
Y
2
Y
3
cob19413_ch02_151-282.qxd 11/22/08 16:37 Page 276
6. For the function g whose graph
is given, find (a) domain,
(b) g(2), and (c) k if
7. The following three points form a right triangle:
( ), (2, 2) and (5, 5). Use the distance formula to
help determine which point is at the vertex of the right
angle. Then find the equation of the smallest circle,
centered at that point, that encloses the triangle.
8. Discuss the end behavior of
F(x) and name the vertex, axis
of symmetry, and all intercepts.
3, 7
g1k23.
1. Write the given equation in slope-intercept form:
2. Find the equation of the line perpendicular to
that passes through (1, 3).
3. Find the implied domain of:
a. b.
4. Given , find
a. b. c.
5. State the equation of the line shown, in slope-
intercept form.
55
5
5
x
y
p1a 12p13a2p a
1
3
b
p1x2x
2
3x 1
g1x2
1
12x 3
f 1x2
x 1
x
2
5x 4
x 2y 8
4x 3y 12
2-127 Mixed Review 277
MIXED REVIEW
•
The composition of two functions is written (g is an input for f ).
•
The domain of is all x in the domain of g, such that g(x) is in the domain of f.
•
To evaluate , we find then substitute . Alternatively, we can find , then find f(k).
•
A composite function can be “decomposed” into individual functions by identifying functions f
and g such that . The decomposition is not unique.
EXERCISES
For and find the following:
49. 50. 51. the domain of
Given and find:
52. 53.
54. and
For each function here, find functions f(x) and g(x) such that
55. 56.
57. A stone is thrown into a pond causing a circular ripple to move outward from the point of entry. The radius of the
circle is modeled by where t is the time in seconds. Find a function that will give the area of the
circle directly as a function of time. In other words, find
58. Use the graph given to find the value of each expression:
a.
b.
c.
d.
e. 1f
#
g2132
a
g
f
b1102
1g f 2172
1g f 2152
1f g2122
1052
2
8
x
y
g(x)
(2, 5)
(2, 1)
f(x)
A1t2.
r1t2 2t 3,
h1x2 x
2
3
3x
1
3
10h1x2 13x 2 1
h1x2 f 3g1x24:
1r p21x21p r21x21q p2132
1p q21x2
r 1x2
x 3
4
q1x2 x
2
2x,p1x2 4x 3,
a
f
g
b1x21f
#
g21321f g21a2
g1x2 3x 2,f1x2 x
2
4x
1f g21x2 h1x2
h1x2 1f g21x2
g122 kx 21f g21x21f g2122
f g
1f g21x2 f 3g1x24
55
5
5
x
y
F(x)
55
5
5
x
y
g(x)
College Algebra—
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9. Graph by plotting the y-intercept, then counting
to find additional points:
10. Solve the inequality using the graph provided:
; .
11. a. Graph the function By
observing the graph, is the average rate of change
positive or negative in the interval ?
Why? Do you expect the rate of change in [1, 2]
to be greater or less than the rate of change in
? Calculate the average rate of change
in each interval and comment.
b. If $1000 is deposited in an account paying 7%
interest compounded continuously, the function
model is . Use the average rate
of change formula to determine if the amount of
interest added to the account exceeds $200 per
year in the 10th, 15th, or 20th year.
Use the intervals [10, 10.01], [15, 15.01], and
[20, 20.01].
Given and , find
12.
13. and its domain
14. Sketch the function h as defined.
15. Given and , calculate
the difference quotient for each function and use
the results to estimate the value of x for which their
rates of change are equal.
16. Identify the function family for the function
. Then sketch the graph using
transformations of a parent function and a few
characteristic points.
g1x22
x 3
4
g1x2 3x 2f 1x2 x
2
1
h1x2 •
50 x 6 8
x 38 x 15
2x 40 x 7 15
1f g21x2
g
f
a
1
2
b
g1x2 3x 2f 1x2
3
x
2
1
a
¢A
¢t
7 200b
A1t2 1000e
0.07t
32, 14
32, 14
p1x22x
2
8x.
55
5
5
x
y
f 1x26 0f 1x2 4x
4
3
x
2
y
3
5
x 2
m
¢y
¢x
17. For the graph shown, determine
a. the domain and range of g,
b. intervals where g is increasing, decreasing, or
constant,
c. intervals where g is positive or negative,
d. any maximum or minimum values for g.
18. Draw a function f that has the following characteris-
tics, then write the zeroes of the function and the
location of all maximum and minimum values.
a. domain:
b. range:
c.
d.
e.
f. for
g. for
h. for
19. Find the equation of the function f(x) whose graph
is given.
20. Since 1975, the number of deaths in the United
States due to heart disease has been declining at a
rate that is close to linear. Find an equation model
if there were 431 thousand deaths in 1975 and
257 thousand deaths in 2000 (let x represent years
since 1975 and f(x) deaths in thousands). How many
deaths due to heart disease does the model predict
for 2008?
Source: 2004 Statistical Abstract of the United States,
Table 102
5
(1.5, 0)
(0, 3)
(1, 0)
5
5
5
x
y
x 320, 304f 1x2 5
x 10, 22 ´ 110, 302f 1x27 0
x 12, 102f1x26 0
f 1x2c for x 15, 152
f 1x2T from x 10, 52 ´ 115, 202
f 122 f 1102 0
y 310, 124
x 3
0, 304
1010
10
10
x
y
g(x)
278 CHAPTER 2 Relations, Functions, and Graphs 2-128
College Algebra—
cob19413_ch02_151-282.qxd 11/22/08 16:40 Page 278
2-129 Practice Test 279
PRACTICE TEST
1. Two relations here are functions and two are not.
Identify the nonfunctions (justify your response).
a. b.
c. d.
2. Determine if the lines are parallel, perpendicular,
or neither:
L
1
: and L
2
:
3. Graph the line using the slope and y-intercept:
4. Find the center and radius of the circle defined by
, then sketch its graph.
5. Find the equation of the line parallel to ,
containing the point (2, ). Answer in slope-
intercept form.
6. My partner and I are at coordinates on a
map. If our destination is at coordinates ,
(a) what are the coordinates of the rest station
located halfway to our destination? (b) How far
away is our destination? Assume that each unit
is 1 mi.
7. Write the equations for lines L
1
and L
2
shown.
8. State the domain and range for the relations shown
on graphs 8(a) and 8(b).
Exercise 8(a) Exercise 8(b)
9. For the linear function shown,
a. Determine the value of
W(24) from the graph.
b. What input h will give an
output of
c. Find a linear function for
the graph.
W1h2 375?
64
5
5
y
xx
y
64
5
5
55
5
5
L
1
L
2
y
x
135, 122
120, 152
2
6x 5y 3
x
2
y
2
4x 6y 3 0
x 4y 8
y
2
5
x 7.2x 5y 15
y x
2
2x
y
1 x
y 15 2x
x y
2
2y
d. What does the slope indicate in this
context?
e. State the domain and range of h.
10. Each function graphed here is from a toolbox
function family. For each graph, (a) identify the
function family, (b) state the domain and range,
(c) identify x- and y-intercepts, (d) discuss the end
behavior, and (e) solve the inequality
and (f) solve
11. Given , evaluate and simplify:
(a) (b) (c)
12. Given and ,
determine and its domain.
13. Monthly sales volume for a successful new
company is modeled by , where
S(t) represents sales volume in thousands in
month t ( corresponds to January 1).
(a) Would you expect the average rate of change
from May to June to be greater than that from
June to July? Why? (b) Calculate the rates of
change in these intervals to verify your answer.
(c) Calculate the difference quotient for S(t) and
use it to estimate the sales volume rate of change
after 10, 18, and 24 months.
Sketch each graph using a transformation.
14.
15.
16. A snowball increases in size as it rolls downhill.
The snowball is roughly spherical with a radius that
can be modeled by the function where tr1t2 1t
,
g1x21x 32
2
2
f 1x2
x 2
3
t 0
S1t2 2t
2
3t
1f g21x2
g1x2 13x 1
f 1x2 x
2
2
f 11 2i2f 1a 32
f 1
2
3
2
f 1x2
2 x
2
x
2
x
y
IV.
55
5
5
x
y
III.
55
5
5
x
y
II.
55
5
5
x
y
I.
55
5
5
f 1x26 0.
f 1x27 0,
4080162432
500
300
400
200
100
Hours worked
Wages earned
W(h)
h
cob19413_ch02_151-282.qxd 11/22/08 16:42 Page 279