MATH 1111 – Lecture Notes
1
Equations and Inequalities
1.1
Basic Equations
Definition 1.1 A linear equation in one variable is an equation that is
equivalent to one of the form
ax + b = 0
where a and b are real numbers and x is a variable.
1. Solve the equation 3x + 2 = 11.
2. Solve the equation 5x − 1 = 3x + 17.
1
3. Solve the equation 2(1 − x) = 3(1 + 2x) + 5.
4. Solve the equation
2
3
=
.
t+6
t−1
5. Solve the equation
4
16
1
+
+
= 0.
3 − t 3 + t 9 − t2
2
Rule 1.2 The power equation xn = a has the solution
√
x = n a if n is odd
√
x = ± n a if n is even and a ≥ 0
If n is even and a < 0, the equation has no real solution.
6. Find all real solutions of the equation x2 − 24 = 0.
7. Find all real solutions of the equation (x + 2)2 = 4.
3
8. Find all real solutions of the equation 64x6 = 27.
9. Find all real solutions of the equation 3(x − 3)3 = 375.
10. Find all real solutions of the equation x4/3 − 16 = 0.
4
1.2
Modeling with Equations
Guidelines for modeling with equations
• Identify the variable
• Translate from words to algebra
• Set up the model
• Solve the equation and check
1. Express the given quantity in terms of the indicated variable. The
average of four quiz scores if each of the first three scores is 8; q =
fourth quiz score.
2. Express the given quantity in terms of the indicated variable. The sum
of three consecutive integers; n = the first integer of the three.
5
3. A car rental company charges $30 a day and 15 cents a mile for renting
a car. Helen rents a car for two days, and her bill comes to $108. How
many miles did she drive?
4. A phone company charges $50 a month and 2 cents a minute for each
minute used over 400 minutes. Angela’s cell phone bill is $54.92. How
many minutes did she use?
6
Rule 1.3 The formula for simple interest is
I = P rt
where I is the total interest earned and P is the principal deposited for t years
at interest rate r.
5. Express the given quantity in terms of the indicated variable. The
interest obtained after one year on an investment at 3% simple interest
per year; x = number of dollars invested.
6. Angie invested $12,000, a portion earning a simple interest rate of 4 21 %
per year and the rest earning a rate of 4% per year. After 1 year the
total interest earned on these investments was $525. How much did she
invest at each rate?
7
Rule 1.4 Distance = rate × time
7. Express the given quantity in terms of the indicated variable. The rate
in miles per hour it takes to travel a given distance in 4 hours; d =given
distance.
8. Bill left his house at 2:00PM and rode his bicycle down Main Street
at a speed of 12 mi/h. When his friend Mary arrived at his house at
2:10PM, Bill’s mother told her the direction in which Bill had gone,
and Mary cycled after him at a speed of 16 mi/h. At what time did
Mary catch up with Bill?
8
9. A square plot of land has a building 60 ft long and 40 ft wide at one
corner. The rest of the land outside the building forms a parking lot.
If the parking lot has area 12,000 ft2 , what are the dimensions of the
entire plot of land?
10. Bill earns $10 an hour at his job, but if he works more than 40 hours
in a week, he is paid 1 21 times his regular salary for the overtime hours.
One week he made $475. How many overtime hours did he work that
week?
9
11. A movie star, unwilling to give his age, posed the following riddle to
a gossip columnist: ”Seven years ago, I was eleven times as old as my
daughter. Now I am four times as old as she is.” How old is the movie
star?
1.3
Quadratic Equations
Definition 1.5 A quadratic equation is an equation of the form
ax2 + bx + c = 0
where a, b, and c are real numbers with a 6= 0.
Rule 1.6 Zero-Product Property
xy = 0 if and only if
10
1. Solve the quadratic equation x2 − 7x + 12 = 0 by factoring.
2. Solve the quadratic equation 3x2 + 1 = 4x by factoring.
2
Rule
2 1.7 Completing the Square To make x + bx a perfect square, add
b
to both sides of your equation. This gives
2
2
b
x + bx +
=
2
2
11
3. Solve the quadratic equation x2 − 4x + 2 = 0 by completing the square.
4. Solve the quadratic equation 2x2 + 8x = −1 by completing the square.
12
5. Solve the quadratic equation x2 − 7x = 5 by completing the square.
6. Solve the quadratic equation 5x2 − 10x + 25 = 0 by completing the
square.
13
Rule 1.8 Quadratic Formula The solutions of the quadratic equation
ax2 + bx + c = 0, where a 6= 0 are
√
−b ± b2 − 4ac
x=
2a
7. Solve the quadratic equation 2x2 − 3 = x by using the quadratic formula.
8. Solve the quadratic equation x2 + 3x + 1 = 0 by using the quadratic
formula.
14
9. Solve the quadratic equation x2 − 2x − 15 = 0.
10. Solve the quadratic equation z 2 − 32 z +
9
16
11. Solve the quadratic equation 3y 2 + y = 2
15
= 0.
Rule 1.9 The discriminant of the the equation ax2 + bx + c = 0 is
D=
If D > 0, the equation has two real solutions. If D < 0, the equation has no
real solution. If D = 0, the equation has exactly one real solution.
12. Use the discriminant to determine the number of real solutions of the
equation x2 = 8x + 5. Do not solve the equation.
13. Use the discriminant to determine the number of real solutions of the
= 0. Do not solve the equation.
equation 4x2 + 5x + 25
2
14. Use the discriminant to determine the number of real solutions of the
equation 9x2 − 6x = −1. Do not solve the equation.
16
1.5
Other Types of Equations
• Polynomial Equations
• Equations Involving Radicals
• Equations of Quadratic Type
• Applications
1. Find all real solutions of the equation x3 = 16x.
2. Find all real solutions of the equation x5 + 8x2 = 0.
17
3. Find all real solutions of the equation (x + 1)5 − 9(x + 1)3 = 0.
4. Find all real solutions of the equation x3 − 5x2 − 2x + 10 = 0.
5. Find all real solutions of the equation
18
1
5
1
+
= .
x−1 x+2
4
6. Find all real solutions of the equation
√
2x + 1 + 1 = x.
Definition 1.10 An equation of quadratic type has the form
aW 2 + bW + c = 0
where a, b, and c are real numbers with a 6= 0.
7. Find all real solutions of the equation x4 − 13x2 + 40 = 0.
19
8. Find all real solutions of the equation x6 − 2x3 − 3 = 0.
√
9. Find all real solutions of the equation x − 4 x − 1 = 0.
20
10. A group of friends decides to buy a vacation home for $120,000, sharing
the cost equally. If they can find one more person to join them, each
person’s contributions will drop by $6,000. How many people are in
the group?
11. A box has volume of 6 ft3 . Its length is 1 ft less than its height, and its
width is 3 ft greater than its height. What are the dimensions of the
box?
21
1.6
Inequalities
Rule 1.11 An inequality looks like an equation where the = is replaced
with <, >, ≤, or ≥. The solution to an inequality is the set of values that
make the inequality true. The solutions looks like an inequality or set of
inequalities with the variable isolated on one side.
√
1. Let S = {−2, −1, 0, 21 , 1, 2, 2, 4}. Determine which elements of S
satisfy the inequality 1 < 2x − 4 ≤ 7.
√
2. Let S = {−2, −1, 0, 21 , 1, 2, 2, 4}. Determine which elements of S
satisfy the inequality x2 + 2 < 4.
22
Rule 1.12
• Addition and Subtraction Adding or subtraction a given
quantity on each side of an inequality gives an equivalent inequality.
• Multiplication and Division with a POSITIVE value Multiplying or dividing by a positive quantity on each side of an inequality
gives an equivalent inequality.
• Multiplication and Division with a NEGATIVE value Multiplying or dividing by a negative quantity on each side of an inequality
reverses the direction of the inequality.
3. Solve the linear inequality 2x − 5 > 3. Express the solution using
interval notation and graph the solution set.
4. Solve the linear inequality 3x + 11 ≤ 6x + 8. Express the solution using
interval notation and graph the solution set.
23
5. Solve the linear inequality 4 − 3x ≤ −1(1 + 8x). Express the solution
using interval notation and graph the solution set.
6. Solve the linear inequality −1 < 2x − 5 < 7. Express the solution using
interval notation and graph the solution set.
≥ 61 . Express the solution using
7. Solve the linear inequality 23 ≥ 2x−3
12
interval notation and graph the solution set.
24
Guidelines for solving nonlinear inequalities
• Move all terms to one side
• Factor
• Find the intervals
• Make a table or diagram
• Solve
8. Solve the nonlinear inequality (x − 5)(x + 4) ≥ 0. Express the solution
using interval notation and graph the solution set.
9. Solve the nonlinear inequality x2 − 3x − 18 ≤ 0. Express the solution
using interval notation and graph the solution set.
25
10. Solve the nonlinear inequality 3x2 − 3x < 2x2 + 4. Express the solution
using interval notation and graph the solution set.
11. Solve the nonlinear inequality x3 − 16x < 0. Express the solution using
interval notation and graph the solution set.
26
12. Solve the nonlinear inequality (2x − 1)(−x + 3) ≥ 0. Express the
solution using interval notation and graph the solution set.
1.7
Absolute Value Equations and Inequalities
Rule 1.13
|x| = C is equivalent to
1. Solve the equation |4x| = 24.
27
2. Solve the equation 5|x| + 3 = 28.
3. Solve the equation |x + 4| = 12 .
4. Solve the equation 8 + 5| 31 x − 56 | = 33.
28
5. Solve the equation |x − 1| = |3x + 2|.
Rule 1.14 Properties of Absolute Value Inequalities
Inequality
|x| < c
|x| ≤ c
|x| > c
|x| ≥ c
Equivalent Form
−c < x < c
−c ≤ x ≤ c
x < −c and c < x
x ≤ −c and c ≤ x
6. Solve the inequality |x − 5| ≤ 3. Express the answer using interval
notation.
29
| ≥ 4. Express the answer using interval nota7. Solve the inequality | x+1
2
tion.
8. Solve the inequality 2| 21 x + 3| + 3 ≤ 51. Express the answer using
interval notation.
30
9. Solve the inequality 0 < |x − 5| < 2. Express the answer using interval
notation.
10. Solve the inequality 0 < |2x − 6| < 10. Express the answer using
interval notation.
31
1
| > 2. Express the answer using interval nota11. Solve the inequality | x+7
tion.
4
12. Solve the inequality | 2x−5
| ≥ 1. Express the answer using interval
notation.
32
1
| < 1. Express the answer using interval nota13. Solve the inequality | x−9
tion.
2
Coordinates and Graphs
2.1
The Coordinate Plane
Vocabulary
• Cartesian plane
• x-axis
• y-axis
• quadrants
• origin
• ordered pair
• x-coordinate
• y-coordinate
33
1. Plot the points (2, 3), (−2, 3), (4, 5), (4, −5), (−4, 5), (−4, −5) in the coordinate plane.
2. Sketch the region given by the set {(x, y)|x = 3}.
3. Sketch the region given by the set {(x, y)|1 < y < 2}.
34
4. Sketch the region given by the set {(x, y)|x ≥ 1 and y < 3}.
5. Sketch the region given by the set {(x, y)|x < −2 and y < 5}.
Theorem 2.1 The Distance Formula The distance between the points
A(x1 , y1 ) and B(x2 , y2 ) is
D(A, B) =
35
Theorem 2.2 The Midpoint Formula The midpoint of the line segment
that joins the points A(x1 , y1 ) and B(x2 , y2 ) is
x1 + x2 y1 + y2
,
2
2
6. Plot the points (−3, −6), (4, 18) in the coordinate plane. Find the
distance between them. Find the midpoint of the line segment that
joints them.
7. Plot the points (0, 8), (6, 16) in the coordinate plane. Find the distance
between them. Find the midpoint of the line segment that joints them.
36
8. Plot the points (5, 0), (0, 6) in the coordinate plane. Find the distance
between them. Find the midpoint of the line segment that joints them.
9. Plot the points (6, −2), (−1, 3) in the coordinate plane. Find the distance between them. Find the midpoint of the line segment that joints
them.
37
10. Which of the points (4, 9), (−7, 6) is closer to the origin?
11. Which of the points (−6, 3), (3, 0) is closer to the point (−2, 1)?
38
12. Graph the triangle with vertices (6, −7), (11, −3), (2, −2). Find its area.
What else can you say about this triangle?
2.2
Graphs of Equations in Two Variables
Definition 2.3 The graph of an equation in variables x and y is the set of
all points in the Cartesian plane (x, y) that satisfy the equation.
1. Determine whether the points (0, 0), (0, 1), (−1, −1) are on the graph
of the equation x − 2y − 1 = 0.
39
Definition 2.4 The x-intercepts are the x-coordinates of the points where
the graph of an equation intersects the x-axis. The y-intercepts are the ycoordinates of the points where the graph of an equation intersects the y-axis.
How to find intercepts
2. Make a table of values and sketch the graph of the equation 2x − y = 6.
Find the x- and y- intercepts.
40
3. Make a table of values and sketch the graph of the equation y = x2 − 9.
Find the x- and y- intercepts.
4. Make a table of values and sketch the graph of the equation x2 +y 2 = 25.
Find the x- and y- intercepts.
41
5. Find the x- and y-intercepts of the equation x2 − xy + y = 1.
6. Find the x- and y-intercepts of the equation xy = 5.
42
Definition 2.5 The standard form of the equation of a circle with center
(h, k) and radius r is
(x − h)2 + (y − k)2 = r2 .
7. Find the center and radius of the circle (x − 3)2 + (y − 4)2 = 49.
8. Find the center and radius of the circle x2 + (y + 6)2 = 81.
43
9. Find the equation of the circle with center (1, −2) containing the point
(1, 1).
10. Find the equation of the circle that satisfies the condition that the
endpoints of a diameter are P (−1, 1) and Q(5, 9).
44
11. Show the equation x2 + y 2 − 4x + 10y + 13 = 0 represents a circle, and
find the center and radius of the circle.
12. Show the equation x2 + y 2 + x = 0 represents a circle, and find the
center and radius of the circle.
45
Definition 2.6 A graph is symmetric about the x-axis if the graph is
unchanged when reflected about the x-axis. A graph is symmetric about
the y-axis if the graph is unchanged when reflected about the y-axis. A
graph is symmetric about the origin if the graph is unchanged when
rotated 180◦ about the origin.
How to test for symmetry
46
13. Test the equation for symmetry.
(a) y = x4 + x2
(b) y = x3 + 10x
(c) y = x2 + 1
(d) x + y 4 − y 2
47
2.4
Lines
Definition 2.7 The slope of a line that passes through the points A(x1 , y1 )
and B(x2 , y2 ) is
m=
1. Find the slope of the line through P (1, −3) and Q(−1, 6).
2. Find the slope of the line through P (−1, −4) and Q(5, 0).
Definition 2.8 The Point-Slope Equation of a Line that passes through
the point (x1 , y1 ) and has slope m is
y − y1 = m(x − x1 ).
48
The Slope-Intercept Equation of a Line that has slope m and y-intercept
b is
y=
The Equation of a Vertical Line through the point (a, b) is
x=
The Equation of a Horizontal Line through the point (a, b) is
y=
Definition 2.9 Two lines with slopes m1 and m2 are parallel if
m1 =
Two lines with slopes m1 and m2 are perpendicular if
m1 =
Also a horizontal line is perpendicular to a vertical line.
3. Find an equation of the line that satisfies the given conditions. Through
(−2, 4); slope −1.
49
4. Find an equation of the line that satisfies the given conditions. Through
(−1, 2) and (4, 3).
5. Find an equation of the line that satisfies the given conditions. xintercept 1; y-intercept -3.
6. Find an equation of the line that satisfies the given conditions. Through
(2, −1) and parallel to the line x = 5.
50
7. Find an equation of the line that satisfies the given conditions. Through
(4, 3); parallel to the x-axis.
8. Find an equation of the line that satisfies the given conditions. Through
( 12 , − 32 ); perpendicular to the line 4x − 8y = 1.
9. Find an equation of the line that satisfies the given conditions. xintercept 0; parallel to the line 2x − y = 6.
51
10. Find the slope and y-intercept of the line 3x − 2y = 12 and draw its
graph.
11. Find the slope and y-intercept of the line x = 9 and draw its graph.
52
3
Functions
3.1
What Is a Function?
that assigns to each eleDefinition 3.1 A function f is a
ment x in a set A exactly one element, called f (x), in a set B.
1. Example
(a) Rule
(b) Machine
(c) Arrow Diagram
Vocabulary
• image of x under f
• domain
• range
• independent variable
• dependent variable
• input
• output
53
There are four ways to represent a function.
• V
• A
• G
• N
2. Express the rule, “Subtract 5, then square” in function notation.
3. Express the function g(x) =
x
3
− 4 verbally and visually.
54
4. Express the function f (x) = 2(x − 1)2 numerically using x-values from
the set {−1, 0, 1, 2, 3}.
5. Evaluate the piecewise defined function at the values
f (−2), f (−1), f (0), f (1).
2
x
if x < 0
f (x) =
x + 1 if x ≥ 0
6. Evaluate the piecewise defined function at the values
f (−4), f (− 32 ), f (−1), f (0), f (25).
 2
 x + 2x if x ≤ −1
x
if − 1 < x ≤ 1
f (x) =

−1
if x > 1
55
Definition 3.2 The domain of a function is the set of real numbers on
which the function is
.
7. Find the domain of the function g(x) = 2x + 1.
8. Find the domain of the function h(x) = 3x on 0 ≤ x < 5.
56
9. Find the domain of the function f (x) =
10. Find the domain of the function h(x) =
11. Find the domain of the function g(x) =
57
x2
1
.
+x−6
3
.
x+1
√
x − 5.
12. Find the domain of the function f (x) = √
1
.
3−x
√
4−x
.
13. Find the domain of the function g(x) =
x+7
√
14. Find the domain of the function f (x) =
58
2+x
.
3−x
3.2
Graphs of Functions
1. Sketch the graph of the function f (x) = 2x − 4 by first making a table
of values.
2. Sketch the graph of the function g(x) = 16 − x2 by first making a table
of values.
59
3. Sketch the graph of the function h(x) = |x + 1| by first making a table
of values.
√
4. Sketch the graph of the function g(x) = 1 + x by first making a table
of values.
60
Basic graphs of some functions
• Linear functions
• Power functions
• Root functions
• Reciprocal functions
• Absolute value functions
61
5. Sketch the graph of the piecewise defined function.
3
if x < 2
f (x) =
x − 1 if x ≥ 2
6. Sketch the graph of the piecewise defined function.
2
x
if x ≤ 0
f (x) =
1
if x > 0
62
7. Sketch the graph of the piecewise defined function.
2x + 3 if x < −1
f (x) =
3−x
if x ≥ −1
Rule 3.3 The Vertical Line Test
A curve in the coordinate plane is the graph of a function if and only if no verthe curve more than
tical line
63
.
8. Use the Vertical Line Test to determine whether the curve is the graph
of a function of x. If it is, give the domain and range of the function.
9. Use the Vertical Line Test to determine whether the curve is the graph
of a function of x. If it is, give the domain and range of the function.
64
10. Use the Vertical Line Test to determine whether the curve is the graph
of a function of x. If it is, give the domain and range of the function.
11. Does the equation define y as a function of x?
(a) y − x2 = 2
(b) y 2 − x = 2
65
3.3
Getting Information from the Graph of a Function
1. The graph of a function f is given below.
(a) Find f (−1), f (1), f (4).
(b) Find the domain and range of f .
(c) For which values of x is f (x) = 0?
2. The graph of a function f is given below.
(a) Find f (0), f (2), f (−2).
(b) Find the domain and range of f .
(c) For which values of x is f (x) = 3?
66
3. Graphs of the functions f and g are given below.
(a) Which is larger g(1) or f (1)?
(b) For which values of x is f (x) = g(x)?
Definition 3.4
whenever
• f is increasing on an interval I if
in I.
• f is decreasing on an interval I if
in I.
67
whenever
4. Determine the intervals on which the function given below is increasing
and decreasing.
5. Determine the intervals on which the function given below is increasing
and decreasing.
68
Definition 3.5
• The function value f (a) is a local maximum value
of f if
when x is near a. In this case we say f has a
local maximum at x = a.
• The function value f (b) is a local minimum value of f if
when x is near b. In this case we say f has a local minimum at x = b.
6. Determine the intervals on which the function given below is increasing
and decreasing. Final all local maximum and minimum values of the
function and the value of x at which each occurs.
69
7. Determine the intervals on which the function given below is increasing
and decreasing. Final all local maximum and minimum values of the
function and the value of x at which each occurs.
8. The graph shows the speed of a car commuting to work. Determine
the intervals on which the function is increasing and decreasing. What
happened between 7:15 and 7:35? What happened at 7:10?
70
9. Three runners compete in a 100-meter hurdle race. The graph depicts
the distance run as a function of time for each runner. What does this
graph tell you about the race? Who won? Did each runner finish?
What happened to runner B?
3.4
Average Rate of Change of a Function
Definition 3.6 The average rate of change of a function y = f (x) between x = a and x = b is
average rate of change =
The average rate of change is the slope of the
x = a and x = b.
71
f (b) − f (a)
.
b−a
line between
1. The graph of a function is given. Determine the average rate of change
of the function between the indicated points on the graph.
2. The graph of a function is given. Determine the average rate of change
of the function between the indicated points on the graph.
72
3. Determine the average rate of change of the function f (x) = x2 + 2x
between x = −1 and x = 4.
4. Determine the average rate of change of the function h(x) =
tween x = 4 and x = 16.
√
5. Determine the average rate of change of the function f (x) =
between x = −0 and x = 3.
73
x be-
2
x+1
6. The table shows the number of CD players sold in a small electronics
store in the years 1993-2003.
Y ear CD players sold
1993
512
1994
520
1995
413
1996
410
1997
468
1998
510
1999
590
2000
607
2001
732
2002
612
2003
584
(a) What was the average rate of change of sales between 1993 and
2003?
(b) What was the average rate of change of sales between 1994 and
1996?
(c) Between which two successive years did sales increase most quickly?
Decrease most quickly?
74
3.5
Transformations of Functions
Vertical Shifts of Graphs
Suppose c > 0
• To graph y = f (x) + c, shift the graph of y = f (x)
c units.
• To graph y = f (x) − c, shift the graph of y = f (x)
c units.
75
Horizontal Shifts of Graphs
Suppose c > 0
• To graph y = f (x+c), shift the graph of y = f (x) to the
c units.
• To graph y = f (x−c), shift the graph of y = f (x) to the
c units.
76
Reflecting Graphs
• To graph y = −f (x), reflect the graph of y = f (x) in the
.
• To graph y = f (−x), reflect the graph of y = f (x) in the
.
77
Vertical Stretching and Shrinking of Graphs
To graph y = cf (x):
• If c > 1,
of c.
• If 0 < c < 1,
factor of c.
the graph of y = f (x) vertically by a factor
the graph of y = f (x) vertically by a
78
Horizontal Stretching and Shrinking of Graphs
To graph y = f (cx):
• If c > 1,
of 1/c.
• If 0 < c < 1,
factor of 1/c.
the graph of y = f (x) horizontally by a factor
the graph of y = f (x) horizontally by a
79
1. Sketch the graph of the function f (x) = |x − 1| + 5, not by plotting
points, but by starting with the graph of a standard function and applying transformations.
√
2. Sketch the graph of the function f (x) = −x + 3, not by plotting
points, but by starting with the graph of a standard function and applying transformations.
80
3. Sketch the graph of the function f (x) = −(x + 1)2 , not by plotting
points, but by starting with the graph of a standard function and applying transformations.
4. Sketch the graph of the function f (x) = 4x3 −1, not by plotting points,
but by starting with the graph of a standard function and applying
transformations.
81
5. A function f is given, and the indicated transformations are applied
to its graph in the given order. Write and equation for the final transformed graph.
f (x) = |x|; shift 3 units to the right and shift upward 1 unit
6. A function f is given, and the indicated transformations are applied
to its graph in the given order. Write and equation for the final transformed graph.
f (x) = x2 ; stretch vertically by a factor of 2, shift downward 2 units,
and shift 3 units to the right
82
7. The graphs of f and g are given. Find a formula for the function g.
8. The graphs of f and g are given. Find a formula for the function g.
83
3.6
Combining Functions
Let f and g be functions with domains A and B. Then
(f + g)(x) =
Domain
(f − g)(x) =
Domain
(f g)(x) =
Domain
f
(x) =
g
Domain
1. Find f +g, f√−g, f g, and f /g and their domains where f (x) =
and g(x) = 1 − x.
84
√
4 − x2
2
2. Find f + g, f − g, f g, and f /g and their domains where f (x) =
x+1
x
and g(x) =
.
x+1
Definition 3.7 Given two function f and g, the composition of f and g,
written f ◦ g is defined by
(f ◦ g)(x) =
.
The domain of f ◦ g is all x that are defined for both g(x) and f (g(x)).
85
3. Use f (x) = 6x + 1 and g(x) = x2 − 3 to evaluate the expression.
(a) f (g(0))
(b) g(f (0))
(c) f (f (−1))
(d) g(g(2))
(e) (f ◦ g)(3)
(f) (g ◦ f )(2)
4. Use the graphs of f and g to evaluate the expresssion.
(a) f (g(0))
(b) g(f (0))
(c) f (f (−1))
(d) g(g(2))
(e) (f ◦ g)(3)
(f) (g ◦ f )(2)
86
5. Find the functions f ◦ g, g ◦ f , f ◦ f , and g ◦ g and their domains where
f (x) = 6x − 5 and g(x) = x2 .
87
6. Find the functions f ◦ g,
√ g ◦ f , f ◦ f , and g ◦ g and their domains where
2
f (x) = x and g(x) = x − 1.
88
7. Find the functions f ◦ g, g ◦ f , f ◦ f , and g ◦ g and their domains where
1
x
and g(x) = .
f (x) =
x+1
x
89
8. Express the function F (x) = (x − 9)5 in the form f ◦ g.
9. Express the function F (x) =
10. Express the function F (x) =
√
x + 1 in the form f ◦ g.
1
in the form f ◦ g.
x+3
90
3.7
One-to-One Functions and Their Inverses
Definition 3.8 A function with domain A is called a one-to-one function
if f (x1 ) 6= f (x2 ) whenever x1 6= x2 .
Horizontal Line Test
1. Use the graph to determine if f is one-to-one.
91
2. Use the graph to determine if f is one-to-one.
3. Use the graph to determine if f is one-to-one.
92
4. Determine whether the functionf (x) = x4 + 5 is one-to-one.
5. Determine whether the functionf (x) =
93
1
is one-to-one.
x
Definition 3.9 Let f be a one-to-one function with domain A and range
B. Then its inverse function f −1 has domain
and range
and is defined by
f −1 (y) = x ⇐⇒
How to find the inverse of a one-to-one function:
Algebraically
1.
2.
3.
4.
Graphically
1.
94
6. Find the inverse function of f (x) = 4x + 7.
7. Find the inverse function of g(x) = 5 − 4x3 .
95
8. Find the inverse function of h(x) = 1 +
9. Find the inverse function of f (x) =
96
√
1 + x.
2x + 5
.
x−7
10. Find the inverse function of f (x) =
1 − 3x
.
4 + 2x
11. Find the inverse function of f (x) =
4x + 5
.
x−7
97
Rule 3.10 Inverse Function Property
If f and g are inverses, then f (g(x)) = x and g(f (x)) = x.
12. Use the Inverse Function Property to show that f (x) = 3x + 6 and
x−6
are inverses of each other.
g(x) =
3
13. Use the Inverse Function Property to show that f (x) =
g(x) = x3 + 7 are inverses of each other.
98
√
3
x − 7 and
14. Assume that f is a one-to-one function. If f (x) = 3x − 1, find f −1 (8).
15. Assume that f is a one-to-one function. If f (x) =
99
2+x
, find f −1 (2).
1+x
5
Exponential and Logarithmic Functions
5.1
Exponential Functions
Laws of Exponents (Review)
If s, t, a, and b are real numbers, with a > 0 and b < 0, then
• as · at = as+t
• (as )t = ast
• (ab)s = as bs
• ( ab )s =
• a−s =
as
bs
1
as
= ( a1 )s
1. A frog population of 100 frogs is brought to a zoo. It is estimated that
the frog population triples every year.
(a) Estimate the frog population after 2 years? 3 years?
(b) Find a function the models the number of frogs after t years.
(c) Estimate the frog population after 20 years.
100
Definition 5.1 An exponential function is a function of the form
where the base a is a positive real number not equal to one. The domain of
f is the set of all real numbers.
2. Graph the exponential function f (x) = 2x .
3. Graph the exponential function f (x) = ( 12 )x .
101
4. Graph the exponential functions f (x) = 2x and g(x) = 7x on the same
coordinate plane.
5. Graph the exponential functions f (x) = ( 12 )x and g(x) = ( 15 )x on the
same coordinate plane.
102
6. Find the exponential function f (x) = ax whose graph is given.
7. Find the exponential function f (x) = ax whose graph is given.
103
8. Graph f (x) = 2−x −3 and determine the domain, range, and horizontal
asymptote of f .
9. Graph f (x) = ( 13 )x+5 and determine the domain, range, and horizontal
asymptote of f .
104
Compound Interest
The amount A after t years due to a principal P invested at an annual
interest rate r compounded n times per year is
A(t) =
where A(t)=amount after t years, P =Principal, r=interest rate, n= number
of times interest is compounded per year, and t= time of years.
10. Investing $1000 at an annual rate of 12% compounded annually, semiannually, quarterly, monthly, and daily will yield the following amounts
after 3 years.
(a) Annual compounding:
(b) Semiannual compounding:
(c) Quarterly compounding:
(d) Monthly compounding:
(e) Daily compounding:
105
5.2
The Natural Exponential Function
Definition
n 5.2 The number e is defined as the number that the expression
1
1+
approaches as n approaches infinity. In calculus, this is expressed
n
using limit notation as
1
e = lim (1 + )n .
n→∞
n
Definition 5.3 The natural exponential function is the function
with base e. The domain of f is the set of all real numbers.
1. Graph f (x) = −ex+3 and determine the domain, range, and horizontal
asymptote of f .
106
2. Graph f (x) = ex−2 +4 and determine the domain, range, and horizontal
asymptote of f .
3. The atmospheric pressure p on a balloon or plane decreases with increasing height. This pressure, measured in millimeters of mercury, is
related to the number of kilometers h above sea level by the function
p(h) = 760e−0.345h .
(a) Find the atmospheric pressure at a height of 2 kilometers (over 1
mile).
(b) What is the pressure at a height of 10 kilometers?
107
Continuous Compounding
The amount A after t years due to a principal P invested at an annual
interest rate r compounded continuously is
A(t) = P ert .
4. Compute the amount A that results from investing a principal P of
$1000 at an annual rate r of 12% compounded continuously for a time
t of 3 years.
5. Set the following problems up but do not solve down at this time:
(a) How long will it take for an investment to double in value if it
earns 5% compounded continuously?
(b) How long will it take to triple at this rate?
5.3
Logarithmic Functions
Definition 5.4 The logarithmic function to the base a, where a > 0 and
a 6= 1, is denoted by
y = loga x
(read as y is the logarithm to the base a of x) and is defined by
y = loga x ⇐⇒ x = ay .
The domain of the logarithmic function y = loga x is (0, ∞).
1. Express the equation in logarithmic form.
(a) 4 = log3 81
(b) 1.23 = m
(c) eb = 9
(d) a4 = 24
108
2. Express the equation in exponential form.
(a) −1 = log5 ( 15 )
(b) loga 4 = 5
(c) loge b = −3
(d) log3 5 = c
3. Evaluate the following logarithmic expressions.
(a) log10 1000
(b) log10 0.1
(c) log16 4
(d) log2 16
(e) log3
1
27
109
Properties of Logarithms
• loga 1 = 0
• loga a = 1
• loga ax = x
• aloga x = x
4. Evaluate the following logarithmic expressions.
(a) log3 1
(b) log5 58
(c) log2 2
(d) 4log4 12
(e) log7 7x+1
(f) 9log9 (2−x)
110
Rule 5.5 The domain of the logarithmic function = the range of the exponential function =
and
the range of the logarithmic function = the domain of the exponential func.
tion =
5. Sketch the graph of f (x) = log2 x and g(x) = log5 x.
6. Sketch the graph of f (x) = − log2 x and g(x) = log2 (−x).
111
7. Find the domain and sketch the graph of the function f (x) = 2 +
log5 (x + 1).
8. Find the domain and sketch the graph of the function
g(x) = − log10 (x − 3).
112
9. Find the function of the form y = loga x whose graph is given.
10. Find the function of the form y = loga x whose graph is given.
113
Definition 5.6 If the base of the logarithmic function is the number e, then
we have the natural logarithmic function y = loge x = ln x.
If the base of a logarithmic function is the number 10, then we have the
common logarithm function y = log10 x = log x.
11. Use the definition of the logarithmic function to findx.
(a) log3 (4x − 7) = 2
(b) logx 64 = 2
(c) log7 (5x + 4) = 2
(d) log2 x = 2
(e) e2x = 5
(f) 103x = 6
114
5.4
Laws of Logarithms
In the following properties, M , N , and a are positive real numbers, with
a 6= 1 , and r is any real number.
• The Log of a Product Equals the Sum of the Logs
loga (M N ) =
• The Log of a Quotient Equals the Difference of the Logs
M
=
loga
N
• The Log of a Power Equals the Product of the Power and the Log
loga M r =
1. Evaluate each expression.
(a) log4 2 + log4 32
(b) log2 80 − log2 5
115
(c)
−1
3
log2 8
(d) log3 951
(e) log3
√
27
2. Use the Laws of Logarithms to expand each expression.
(a) log2 (6x)
(b) ln(x(x − 1))
116
(c) log5 (x3 y 6 )
(d) ln
(e) log
ab
√
3c
√
x3 x−1
3x−4
3. Use the Laws of Logarithms to combine each expression.
(a) log3 5 + 5 log3 2
(b) 3 log x + 12 log(x + 1)
117
(c) ln(x + y) + ln(x − y) − 4 ln z
(d) 3 ln s + 12 ln t − 4ln(t2 + 1)
(e)
1
3
log(x + 2)3 + 12 [log x4 − log(x2 − x − 6)2 ]
Change of Base Formula
logb x =
loga x
loga b
118
4. Use the Change of Base Formula and common or natural logarithms
to evaluate each logarithm, correct to five decimal places.
(a) log8 5
(b) log9 20
5.5
Exponential and Logarithmic Equations
Guidelines for solving exponential equations:
• Isolate the exponential expression on one side of the equation.
• Take the logarithm of each side.
• Solve for the variable.
1. Solve the equation 5−2x = 7.
119
2. Solve the equation 3x+4 = 6 .
3. Solve the equation 8e9x = 20.
4. Solve the equation 93−2x = 4.
120
5. Solve the equation 3xex + x2 ex = 0.
6. Solve the equation
50
= 4.
1 + e−x
121
Guidelines for Solving Logarithmic Equations:
• Isolate the logarithmic term on one side of the equation; you might first
need to combine the logarithmic terms.
• Write the equation in exponential form (or raise the base to each side
of the equation).
• Solve for the variable.
7. Solve the equation ln x = 8.
8. Solve the equation log2 (25 − x) = 3.
122
9. Solve the equation 4 + 3 log(2x) = 16.
10. Solve the equation log2 (x2 − x − 2) = 2.
11. Solve the equation log(x + 2) + log(x − 1) = 1.
123
12. Solve the equation log2 x + log2 (x − 3) = 2.
13. Find the inverse function of f when f (x) = 23x .
14. Find the inverse function of f when f (x) = log3 (x − 1).
124
15. A man invests $2,000 at an interest rate of 6% per year compounded
continuously.
(a) What is the amount after 2 years?
(b) How long will it take for the amount to be $10,000?
16. A sum of $5,000 is invested at an interest rate of 5% per year. Find the
time required for the money to double if the interest is compounded
according to the following method.
(a) Semiannual
(b) Continuous
125
17. A sum of $1,000 is invested at an interest rate of 4% per year. Find
the time required for the amount to grow to $4,000 if interest is compounded continuously.
5.6
Modeling with Exponential and Logarithmic Equations
Exponential Growth Model
n(t) = n0 ert
where n(t) is the population at time t, n0 is the initial size of the population,
and r is the relative rate of growth.
126
1. A bacteria initially has 100 bacteria and is observed to double every
1.5 hours.
(a) Find an exponential function n(t) for the number of bacteria after
t hours.
(b) How many bacteria are there after 5 hours?
(c) When will the bacteria count reach 20,000?
127
2. The population of certain species of deer triples every 8 years. This type
of deer was introduced into Georgia 24 years and now the population
is 135.
(a) What was the initial size of the deer population?
(b) What will the population be 3 years from now?
(c) Sketch a graph of the deer population.
128
3. The population of a city has a relative growth rate of 4% per year. The
city council is trying to reduce the growth rate to 3.5%. The population
in 2011 is 135,000. Find the projected population for the year 2021 for
the following conditions.
(a) The relative growth rate remains a 4% per year.
(b) The relative growth rate is decreased to 3.5% per year.
4. The population of the United States was 312,448,000 in 2011 and the
rate of growth was 0.85% per year.
(a) By what year will the population have doubled?
(b) By what year will the population have tripled?
129
5. The graph shows the mosquito population for the next few years. Assume that the population grows exponentially.
(a) What was the initial population?
(b) Find a function that models the population in terms of t years
from now.
(c) What is the projected population after 10 years?
130
Radioactive Decay Model
m(t) = m0 e−rt
where m(t) is the mass at time t, m0 is the initial mass, and r is the rate of
decay.
6. The half-life of Flourine-21 is 4 seconds. Suppose we have a 20 mg
sample.
(a) Find a function m(t) that models the mass remaining after t seconds.
(b) How much of the sample will remain after 7 seconds?
(c) After how long will only 1 mg of the sample remain?
131
7. If a 400 g sample of a radioactive element decays to 300 g in 6 hours,
find the half-life of the element.
8. A human fossil has 65% of the carbon-14 that would be present in a
living human being. If the half-life of carbon-14 is 5,730 years, how old
is the fossil?
132
Newton’s Law of Cooling
T (t) = Ts + D0 e−kt
where T (t) is the temperature at time t, D0 is the initial difference in temperature of an object and it’s surrounding temperature Ts , and k is positive
constant.
9. A turkey is taken from an oven when it’s temperature is 185◦ F and is
placed in a room where the temperature is 70◦ F.
(a) If the temperature of the turkey is 150◦ F after half an hour, what
is it’s temperature after 45 minutes?
(b) When will the turkey cool to 100◦ F?
133
10. Normal body temperature is 98.6◦ F. Immediately after death the body
starts to cool. It has been determined that the constant in Newton’s
Law of Cooling is k = 0.1947 with time measured in hours. Suppose
the surrounding temperature is 60◦ F.
(a) Find a function T (t) that models temperature t hours after death.
(b) If the temperature of the body is now 72◦ F, how long ago was
time of death?
11
11.1
Systems of Equations and Inequalities
Systems of Linear Equations in Two Variables
Definition 11.1 A system of equations is a collection of two or more
equations, each containing one or more variables. A solution to a system
of equations consists of values for the variables that are solutions of each
equation of the system.
134
Steps for Solving by Substitution
• Pick one of the equations and solve for one of the variables in terms of
the remaining variables.
• Substitute into the OTHER equation.
• Solve this equation.
• Back-substitute.
1. Solve
2. Solve
2x + y = 1
3x + 4y = 14
x2 + y 2 = 100
3x − y = 10
135
Elimination Method
• Adjust the coefficients by multiplying one or more of the equations
by appropriate numbers so that the coefficient of one variable in one
equation is the negative of its coefficient in the other equation.
• Add the equations to eliminate one variable, then solve for the remaining variable.
• Back-substitute and and solve for the remaining variable.
3. Solve
4. Solve
3x + 2y = 14
x − 2y = 2
x2 + 3y = 10
3x + 2y = 10
136
Number of solutions of a linear system in two variables
• The system has exactly one solution.
• The system has no solution.
• The system has infinitely many solutions.
5. Graph the linear system to determine if it has one solution, no solution,
or an infinite number of solutions. If there is exactly one solution, use
the graph to find it.
x + 2y = 4
2x + 4y = 8
137
6. Graph the linear system to determine if it has one solution, no solution,
or an infinite number of solutions. If there is exactly one solution, use
the graph to find it.
2x − y = 0
3x + 2y = 7
7. Solve the system or show that is has no solution. If the system has
infinitely many solutions, express them in ordered pair form.
4x + 5y = 10
10x + y = 11
138
8. Solve the system or show that is has no solution. If the system has
infinitely many solutions, express them in ordered pair form.
2x + 4y = 1
−5x − 10y = 3
9. Solve the system or show that is has no solution. If the system has
infinitely many solutions, express them in ordered pair form.
3x − 5y = 3
5x + 5y = 21
139
10. Solve the system or show that is has no solution. If the system has
infinitely many solutions, express them in ordered pair form.
7x − y = 4
−21x + 3y = −12
13
13.2
Sequences and Series
Arithmetic Sequences
Definition 13.1 A sequence is a function f whose domain is the set of
natural numbers. The values f (1), f (2), f (3), . . . are called the terms of the
sequence.
1. Find the first five terms and the 100th term of the sequence defined by
each formula.
(a) an = 2n − 1
140
(b) cn = n2 − 1
(c) rn =
(−1)n
2n
Definition 13.2 An arithmetic sequence is a sequence of the form
a, a + d, a + 2d, a + 3d, . . .
The number a is the first term and d is the common difference of the
sequence. The nth term of an arithmetic sequence is given by
an = a + (n − 1)d
141
2. Find the first six terms, the common difference d, the nth term, and
the 300th term of the arithmetic sequence.
(a) 2, 5, . . .
(b) 13, 7, . . .
(c) Given by a = 1 and d = 4
142
Definition 13.3 For the sequence
a1 , a2 , a3 , . . . , an , . . .
the partial sums are
S1 = a1
S2 = a1 + a2
S3 = a1 + a2 + a3
..
.
Sn = a1 + a2 + a3 + · · · + an
..
.
S1 is called the first partial sum, S2 is the second partial sum, and so
on. Sn is called the nth partial sum. The sequence S1 , S2 , S3 , . . . is called
the sequence of partial sums.
3. Find S1 , S2 , and S3 for the sequences in Problem 1.
143
Rule 13.4 Partials Sums of an Arithmetic Sequence are given by the following formulas.
n
(2a + (n − 1)d) or
2
a + an
= n
2
Sn =
Sn
4. Find S3 and Sn for the sequences in Problem 2.
144
5. Find the sum.
(a) 3 + 8 + 13 + · · · + 278
(b) −10 − 9.9 − 9.8 − · · · − 0.1
13.3
Geomentric Sequences
Definition: A geometric sequence is a sequence of the form
a, ar, ar2 , ar3 , . . .
The number a is the first term, and r is the common ratio of the sequence.
The nth term of a geometric sequence is given by
an = arn−1
145
1. Find the first five terms, the common ration r, the nth term, and the
250th term of the geometric sequence.
(a) 5, 15, 45, . . .
(b) 1, 31 , 19 , . . .
(c) Given by a = 2 and r = −5
146
Rule 13.5 Partial Sums of a Geometric Sequence are given by the formula
Sn = a
1 − rn
.
1−r
2. Find S3 and Sn for the sequences in Example 1.
147
3. Find the sum.
(a) 2 + 8 + 32 + 128 + · · · + 2, 097, 152
(b) 1 − 21 + 14 − 81 + · · · −
1
512
148
4. Determine if the following sequences are arithmetic, geometric, or neither. Find the next term of the sequence.
√ √ √ √
(a) 2, 2 2, 3 2, 4 2, . . .
(b) 1, − 32 , 2, − 52 , . . .
(c) 3, 1, 13 , 19 , . . .
(d) 2, 2.75, 3.5, 4.25, . . .
(e)
3 1 1 2
, , , ,...
4 2 3 9
149