College Algebra
MATH 1111
1
Unit 1 – Functions and Linear Functions
o
Sets
o
Linear Equations in One Variable
o
Linear Inequalities in One Variable
o
Linear Equations in Two Variables
o
Linear Systems
o
Relations vs Functions
2
Sets
3
What is a set?
Definition: A set is a collection of objects whose contents can be clearly
determined.
◦ Objects in the set are referred to as elements.
◦ Sets must be well-defined. That is, it’s contents can be clearly determined.
Sets may be represented in different ways.
◦ Word description: ‘Set 𝑁 is the set of whole numbers greater than 0 and less than or equal
to 9.’
◦ Roster method: 𝑁 = {1, 2, 3, 4, 5, 6, 7, 8, 9}
◦ Set Builder Notation: 𝑁 = 𝑥 𝑥 is an integer greater than 0 and less than or equal to 9}
4
The Universal Set
Definition: The universal set is a general set that contains all the elements
under discussion.
Example: The students in class right now.
Example: {0,1,2,3,4,5,6,7,8,9}
5
Subsets
Example: Let 𝐴 = {1, 3, 5, 7} and 𝐵 = {1, 3, 5, 7, 9, 11 }
Then Set 𝐴 is a subset of Set 𝐵.
Thinking back to the example on the last slide, if we let our
universal set be the students in class right now, one subset would
be the students in class right now currently wearing glasses.
6
The Empty Set
Examples:
1. Set of all numbers less than 4 and greater than 10
2. {x | x is a person taller than 12 feet}
7
Operations on Sets - Complement
The complement of a set can also be symbolized as 𝐴𝐶 or 𝐴.
8
Example - Complement
If the universal set is U = {x|x is a positive integer less than 12} and
𝑁 = {2, 3, 5, 6, 7, 9}, what is N?
If the universal set is 𝑈 = a,e,i,o,u and 𝑆 = {a,o,u}, what is 𝑆′?
9
Operations on Sets - Intersection
10
Example - Intersection
If 𝑁 = {1, 2, 3, 4, 5, 6, 7, 8, 9} and 𝑀 = {0, 2, 4, 6, 8, 10},
what is 𝑁 ∩ 𝑀?
What is 𝑁 ∩ ∅?
11
Operations on Sets - Union
12
Example – Union
If 𝑁 = {1, 2, 3, 4, 5, 6, 7, 8, 9} and 𝑀 = {0, 2, 4, 6, 8, 10},
what is 𝑁 ∪ 𝑀?
What is 𝑁 ∪ ∅?
13
Example – Set Operations
If 𝑈 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11},
𝑁 = {1, 2, 3, 4, 5, 6, 7, 8, 9}, 𝑀 = {0, 2, 4, 6, 8, 10}, and 𝑃 = {1, 2}
What is 𝑁 ∪ 𝑀 ′ ?
What is 𝑁 ′ ∩ 𝑀′ ?
What is 𝑀 ∪ ∅?
What is 𝑁 ∩ 𝑀 ∪ 𝑃?
14
Interval Notation
Inequality
Set-Builder
Graph
Interval
1≤𝑥≤3
−4 < 𝑥 < 0
𝑥>5
𝑥≤1
𝑥 < 1 or 𝑥 ≥ 5
2 < 𝑥 ≤ 3 or 𝑥 > 10
All real numbers
15
Example – Graphing and Interval
Graph the set 𝑃 = 𝑥 𝑥 < 2 or 𝑥 ≥ 7 and write it in interval notation.
Graph the set 𝑆 = 𝑥 𝑥 > −3 and 𝑥 ≤ 2} and write it in interval notation.
16
Example - Union of Intervals
Given the two sets, find the union and graph it:
𝐴 = (−2, 6] and 𝐵 = (3, 11]
𝐴∪𝐵 =
17
Example - Intersection of Intervals
Given the two sets, find the intersection and graph it:
𝐴 = (−2, 6] and 𝐵 = (3, 11]
𝐴∩𝐵 =
18
Example - Intersection of Intervals (disjoint)
Given the two sets, find the intersection and graph it:
𝐴 = (−∞, 6] and 𝐵 = (11, ∞)
𝐴∩𝐵 =
19
Example - Intersection of Intervals (disjoint)
Given the two sets, find the intersection and graph it:
𝐴 = −∞, 6 ∪ 12, ∞ and 𝐵 = (−2,23)
𝐴∩𝐵 =
20
Linear Equations –
Single Variable
21
Solving Linear Equations
Solving an equation means finding the value(s) that make the equation true when evaluating
with that value. To find this value, we must use algebra rules to isolate the variable on one side
of the equation.
Examples: Solve for 𝑥.
2𝑥 − 7 = 1
5𝑥 + 3 = 6 − 2𝑥
22
Example - Solving Linear Equations
Solve for 𝑥.
2 𝑥 + 3 − 4 𝑥 − 5 = 8 − 5𝑥
5 5𝑥 − 5 − 8 = 6 6𝑥 − 2 − 3
23
Example - Solving Linear Equations
Solve for 𝑥:
0.15 𝑥 + 3 − 0.07 𝑥 − 60 = 7.9
24
Example – Different Cases
Solve for 𝑥.
12𝑥 − 3 𝑥 + 4 = 9𝑥 − 2
12𝑥 − 3 𝑥 + 4 = 9𝑥 − 12
25
Types of Solutions
There are three cases we can see when solving linear equations:
o One solution – Conditional
o No solutions – Inconsistent Equation/Contradiction
o Infinite number of solutions – Identity
26
Linear Applications
The following equation: 𝐴 = 8𝑡 + 45 gives the total amount of money, 𝐴 (in dollars), Timmy has
𝑡 weeks after his birthday. When will Timmy have $149?
27
Linear Applications
Solve:
5 less than 3 times a number is 19. What is the number?
28
Linear Applications
Maggie is 8 years younger than Bart. In 9 years the sum of their ages will be 28.
How old is Maggie now?
29
Linear Applications
A triangular field is constructed so that one side is twice the length of the shortest side
and the third side is 1 ½ times the length of the shortest side. If the perimeter of the
field is 720 feet, what are the lengths of the three sides of the field?
a. If 𝑥 is the length of the shortest side, write an equation you can use to solve this
application.
b. Solve the equation from part a
◦ The shortest side is:
◦ The second longest sides is:
◦ The longest side is:
30
Linear Applications
After a 30% reduction, you purchase a mountain bike for $196. What was the
price of the bike before the reduction?
a. Using 𝑥 for the variable, write an equation you can use to solve this
application.
b. Solve the equation from part a.
31
Linear Inequalities
32
Solving Linear Inequalities
Solving an inequality means finding the set of values that make the inequality true when
evaluating with that value. To find this value, we must use algebra rules to isolate the variable
on one side of the inequality.
A very important difference between linear equations and linear inequalities is that when we
multiply or divide both sides of an inequality by a negative number, we must flip the
inequality sign.
You will need to write the solution set using one or more of the following methods:
o
Write the solution set as an inequality
o
Write the solution set as an interval
o
Graph the solution set
33
Examples - Linear Inequalities
Solve for 𝑥. Write the solution set as an inequality, graph the solution set, and write the
solution set as an interval.
2𝑥 − 7 > 1
− 5𝑥 + 3 ≤ 6 − 2𝑥
34
Example - Linear Inequalities
Solve for 𝑥. Write the solution set as an inequality, graph the solution set, and write the
solution set as an interval.
30 − 6𝑥 < −3(5 + 7𝑥)
6 𝑥 − 4 + 7 ≤ −5
35
Example - Compound Inequalities
Compound inequalities can be solved by separating both sides of the inequality, or by solving
both inequalities simultaneously.
1 < 3𝑥 + 7 ≤ 22
1 ≤ −2𝑥 + 3 < 15
2𝑥 + 6 ≤ 𝑥 + 6 < 3𝑥 + 12
36
Application – Single Inequality
A local salesman receives a base salary of $1500 a month. He also receives a commission of 8%
on all sales over $500. His goal is to have a monthly income of at least $2500.
a)
Write an inequality you can use to solve for how much the salesman will have to sell each
month to reach his goal.
b) Use your inequality from part (a) to find the minimum amount the salesman would have to
sell each month to reach his goal.
37
Application – Single Inequality
Harry has received scores of 72 and 78 on his first two 100 point tests. What score must he get
on his third 100 point test to have an average of 80 or greater?
38
Linear Equations – Two
Variables
39
Graphing on the Plane
Graphing points involves finding the intersection where the 𝑥- and 𝑦- coordinates meet. Points
are given as ordered pairs 𝑥, 𝑦 .
The plane has two axes, an 𝑥-axis and a 𝑦-axis. The 𝑥axis is horizontal while the 𝑦-axis is vertical. Where the
two axes meet is called the origin and it has
coordinates (0,0).
40
Example – Graphing Points
Plot the following points on the graph:
◦ (3, 5)
◦ (-2, -1)
◦ (6, 0)
◦ (-4, 5)
◦ (1, -3)
◦ (0, 2)
41
Graphing Linear Functions
Graphs of linear functions are straight lines.
42
Slope
The slope of a function can be considered the average rate of change between two points. For
linear functions, the slope is always the same and can be calculated with the following formula.
𝑚=
change in output
=
change in input
∆𝑦
∆𝑥
=
𝑦2 −𝑦1
𝑥2 −𝑥1
rise
= run , 𝑥1 , 𝑦1 and 𝑥2 , 𝑦2 are two points on the line.
Example: Find the slope of the line.
43
Example – Slope
Find the slope of the line that passes through the points: (3,6) and (4,8).
Find the slope of the line that passes through the points: (−3, −7) and (−9,1).
44
Example – Slope
Find the slope of the line that passes through the points: (−3,5) and (−3,8).
Find the slope of the line that passes through the points: (3,5) and (8,5).
45
Comparing Slopes
https://www.desmos.com/calculator/ewukrupt9w?tour=sliders
46
Slope as an Average Rate of Change
What is the average rate of change in a
child’s height between ages 2 and 6?
What is the average rate of change of a
child’s height between ages 13 and 16?
47
Application – Slope
The population of a town has declined. In 1990 it was 60,000. In 2000, it was
58,000.
What is the rate of change of the population?
48
Writing Linear Equations
Linear Equations are typically written in slope-intercept form.
Slope-Intercept Form:
𝒚 = 𝒎𝒙 + 𝒃
The slope is 𝒎 and the 𝑦-intercept is 𝒃.
In applications, the 𝑦-intercept is the initial value.
Point-Slope Form:
𝒚 − 𝒚𝟏 = 𝒎 𝒙 − 𝒙𝟏
49
VUX HOY
50
Example – Find the Slope
Find the slope of the following lines.
𝑥 + 3𝑦 = 33
−6𝑥 − 4𝑦 = 25
𝑦=2
𝑥 = −1
51
Examples – Equation of a Line
Find the equation of the line with a slope of −3 that contains the point (0, 5).
1
2
Find the equation of the line with a slope of that contains the point (2, -3).
52
Examples – Equations of Lines
Find the equation of the line that passes through the points: (−3,5) and (−3,8).
Find the equation of the line that passes through the points: (3,5) and (8,5).
53
Example – Equation of a Line
Find the equation of the line that goes through the points (4, -8) and (36, -32).
54
Application - Initial Value, Rate of Change
A phone salesperson is paid a salary with commission on every phone sold. The
pay in dollars, 𝑃, is determined by the formula:
𝑃(𝑥) = 15𝑥 + 330
where 𝑥 represents the number of phones sold.
o What is the initial value?
o What does the initial value mean in the context of the problem?
o What is the rate of change?
o What does the rate of change mean in the context of the problem?
55
Application – Rate of Change
At 8:00 AM, the temperature was 64℉. At 4:00 PM, the temperature was 86℉. What was the
average rate of change of temperature between 8:00 AM and 4:00 PM?
56
Graphing Lines
There are three ways to graph a line.
o Table of Values
o Slope-Intercept Form
o 𝑥 − and 𝑦 − Intercepts
57
Graphing from a Table of Values
𝑥 + 𝑦 = 10
Pick a value for 𝑥 and substitute into
the equation to get 𝑦:
𝑥
𝑦
Each point is a solution to the equation
58
Graphing Using Slope and the 𝑦-Intercept
1
𝑦 =− 𝑥+3
2
59
Graphing Using the 𝑥 − and 𝑦 − Intercepts
2𝑥 + 3𝑦 = 6
The 𝑦-intercept is where the graph crosses
the 𝑦 axis so 𝑥 = 0.
The 𝑥-intercept is where the graph crosses
the 𝑥 axis so 𝑦 = 0.
60
Horizontal Lines
Graph the function 𝑦 = 3
61
Vertical Lines
Graph the function 𝑥 = −2
62
Horizontal and Vertical Lines
Find the equation of the horizontal line
that goes through the point (3, −4).
Find the equation of the vertical line
that goes through the point (3, −4).
63
Intercepts of Horizontal and Vertical Lines
Find the 𝑥- and 𝑦- intercepts of each of the following equations.
𝑥=4
𝑦 = −2
64
Parallel and Perpendicular Lines
Parallel lines have the same slope. (𝑚1 = 𝑚2 )
The slopes of perpendicular lines are opposite reciprocals.
(𝑚1 = −
1
)
𝑚2
65
Parallel vs Perpendicular
66
Parallel and Perpendicular Lines
Example: Determine if the following pairs of lines are parallel, perpendicular or neither.
6𝑥 + 𝑦 = 7 and 6𝑥 − 36𝑦 = 14
3
2
2𝑥 + 4𝑦 = 2 and 3𝑦 = − 𝑥 − 6
67
Examples – Parallel and Perpendicular
3
Find the equation of the line parallel to 𝑦 = − 𝑥 − 6 and through the point
5
(5, 4).
Find the equation of the line perpendicular to 2𝑥 + 4𝑦 = 2 and through the
point (3, 4).
68
Application – Writing Linear Equations
Last year, Pinwheel Industries introduced a new toy. It cost $5500 to develop the
toy and $40 to manufacture each toy.
1. Give a linear equation in the form 𝐶 = 𝑚𝑛 + 𝑏 that gives the total cost, 𝐶, to
produce 𝑛 of these toys.
2. What is the total cost to produce 𝑛 = 2600 of these toys?
3. How many toys can be produced with $189500?
69
Application – Solving Linear Equations
A College Algebra student has test scores of 78, 82, 62, and 91, each out of 100
points. The final exam is out of 200 points. What is the lowest grade the
student can get on the final to earn a B (average score of 80) in the course?
70
Application – Finding and Using Equations
An athlete begins normal practice for the next marathon during the evening. He begins running
at 6:00 pm. He finishes his run at 7:30 pm and has run 7.5 miles.
• Write an equation representing the distance, 𝑑, the runner has run over time, 𝑡, in hours since
6:00 pm.
• What was his average speed over the course of the run?
• How many miles did he run in the first 45 minutes?
• If he kept running at the same pace for a total of 4 hours, how many miles will he have run?
71
Linear Systems
72
Systems of Equations
A system of equations is two or more equations with two or more variables.
Solving a system of equations requires solving for all the variables
simultaneously.
The solution to the system must satisfy every equation in the system.
73
Solving a System of Equations
There are 4 methods that can be used to solve a system of equations:
1. Graphing
2. Substitution
3. Elimination (Addition)
4. Matrices (We won’t look at these in this class)
74
Solving a System by Graphing
When graphing a system of linear equations, the point at which they intersect is the solution.
Example: Solve by graphing.
−5𝑥 − 4𝑦 = 2
4𝑥 + 𝑦 = 5
75
Solving by Substitution
To solve with the substitution method, first solve one of the equations for one of
the variables. Then substitute the new expression for that variable into the
second equation and solve. Once you have the value for one variable, substitute
that value into either of the original equations to find the value for the second
variable.
Example: −5𝑥 − 4𝑦 = 2
4𝑥 + 𝑦 = 5
76
Solving by Elimination/Addition
−5𝑥 − 4𝑦 = 2
4𝑥 + 𝑦 = 5
77
Example – Solving Linear Systems
2𝑥 − 3𝑦 = 4
6𝑥 − 9𝑦 = 36
78
Example – Solving Linear Systems
Solve:
2𝑥 − 3𝑦 = 4
6𝑥 − 9𝑦 = 12
79
Types of Solutions
80
Applications Involving Systems of
Equations
Many applications involving systems of equations are ‘mixture’ problems. To solve a mixture
problem, write two equations:
1.
Quantity Equation
2.
Value Equation
It can be helpful to draw a table or chart.
81
Application – Coins
Example: Sue has a combination of nickels and dimes. The total number of coins
is 25 and they total $1.50.
How many of each does she have?
82
Application – Tickets
A movie theater charges $9.00 for adults and $7.00 for senior citizens. On a day
when 325 people paid for admission, the total receipts were $2495. How many
adults and how many seniors were there?
83
Application – Break-Even Point
The cost to manufacture a pair of jeans is 𝑦 = 29𝑥 + 1000. The revenue
equation is 𝑦 = 49𝑥. What is the break-even point?
84
Application – Age
Sam is 7 years older than Dean. In 12 years, the sum of their ages will be 51.
How old are they now?
85
Application – Mixing Nuts
A store sells cashews for $6 per pound and pecans for $2 per pound. How many
pounds of each should be mixed to have 25 pounds of a mixture that would sell
for $3.00 per lb?
86
Application – Interest
Connor has $200,000 to invest. A money market fund pays 6% and a mutual
fund pays 8%. How much should he invest in each account to get an annual
return of $15,000?
87
Application – Solutions
How much of each of a 25% alcohol solution and a 35% alcohol solution must be
combined to get 20 liters of a 32% alcohol solution?
88
Motion problems
When solving motion problems, the “dirt” formula is needed:
distance = rate x time
𝑑 = 𝑟𝑡
89
Application – Motion
With the wind, an airplane travels 1120 miles in seven hours. Against the wind, it
takes eight hours. Find the rate of the plane in still air and the velocity of the
wind.
90
Relations vs Functions
91
What is a Relation?
A relation is a correspondence between two sets. One set is all the possible inputs to a relation
and the second set is all the possible outputs of the relation.
Example:
92
Domain and Range
Domain
Range
93
Function Definition
A relation is a function only if every element in the domain corresponds to only one element in
the range. Are these functions?
94
Function Representations
o
Sets of ordered pairs: {(1,1), (2,4), (3,9)}
o
Algebraically: 𝑓 𝑥 = 𝑥 2
o
Table of Values:
o
Graphically:
x
-2
-1
0
1
2
y
4
1
0
1
4
95
Example – Domain, Range, Function (table)
State the Domain and Range.
x
1
2
3
4
5
y
1
4
9
16
25
Domain:
Range:
Is y a function of x?
96
Example – Domain, Range, Function (table)
State the Domain and Range.
x
1
2
3
-2
-1
y
1
4
9
4
1
Domain:
Range:
Is y a function of x?
97
Example – Domain, Range, Function (table)
State the Domain and Range.
x
1
2
3
-2
1
y
1
4
9
4
2
Domain:
Range:
Is y a function of x?
98
Example – Domain, Range, Function (graph)
What is the domain and range?
Domain:
Range:
99
Vertical Line Test
The Vertical Line Test is a tool to test whether or not a graph represents a function.
If ANY vertical line would intersect the graph in more than one place, the graph does not
represent a function.
If NO vertical line intersects the graph in more than one place, the graph represents a function.
100
Example – Domain, Range, Function (graph)
Is this a function?
101
Example – Domain, Range, Function (graph)
What is the domain and range?
Domain:
Range:
Is this a function?
102
Example – Domain, Range, Function (graph)
What is the domain and range?
Domain:
Range:
Is this a function?
103
Example – Function (graph)
Domain:
Range:
Is this a function?
104
Example – Function (graph)
Domain:
Range:
Is this a function?
105
Evaluate a Function – Graphically
What is 𝑓(−1)?
What is 𝑓(1)?
What is 𝑥 when 𝑓(𝑥) = 0?
106
Evaluate a Function – Algebraically
The value of a function is the output of the function. Outputs are typically 𝑦 values when we
graph functions. The function representation for the output , or 𝑦, is 𝑓(𝑥).
Example: If 𝑓(𝑥) = 𝑥2 – 1
What is 𝑓(3)?
What is 𝑓(−2)?
What is 𝑥 when 𝑓(𝑥) = 8?
107
Graphing with Transformations (vertical shift)
If you know the graph of a function, it can be easy to graph a similar function using
transformations.
Adding to a function moves the graph up.
𝑓(𝑥) = 2𝑥 + 2
𝑓 𝑥 + 3 = 2𝑥 + 2 + 3
= 2𝑥 + 5
𝑓(𝑥) = 2𝑥 + 2
𝑓 𝑥 + 3 = 2𝑥 + 5
This is the same graph, just up 3 from the original.
108
Graphing with Transformations (vertical shift)
If you know the graph of a function, it can be easy to graph a similar function using
transformations.
Subtracting from a function moves the graph down.
𝑓(𝑥) = 2𝑥 + 2
𝑓 𝑥 − 1 = 2𝑥 + 2 − 1
= 2𝑥 + 1
𝑓 𝑥 − 1 = 2𝑥 + 1
This is the same graph, just down 1
from the original.
𝑓(𝑥) = 2𝑥 + 2
109
Graphing with Transformations (horizontal shift)
If you know the graph of a function, it can be easy to graph a similar function using
transformations.
Adding to the 𝑥 in the function moves the graph to the left.
𝑓(𝑥) = 2𝑥 + 2
𝑓(𝑥 + 3) = 2(𝑥 + 3) + 2
= 2𝑥 + 6 + 2
= 2𝑥 + 8
𝑓 𝑥 + 3 = 2𝑥 + 8
𝑓(𝑥) = 2𝑥 + 2
This is the same graph, just moved 3 to the left.
110
Graphing with Transformations (horizontal shift)
If you know the graph of a function, it can be easy to graph a similar function using
transformations.
Subtracting from the 𝑥 in a function moves the graph to the right.
𝑓 𝑥 − 4 = 2𝑥 − 6
𝑓 𝑥 = 2𝑥 + 2
𝑓 𝑥−4 =2 𝑥−4 +2
𝑓(𝑥) = 2𝑥 + 2
= 2𝑥 − 8 + 2
= 2𝑥 − 6
This is the same graph, just moved 4 to the right.
111
Graphing with Transformations (reflection)
If you know the graph of a function, it can be easy to graph a similar function using transformations.
Changing the sign of the function creates a reflection, or mirror image, of the graph across the 𝑥axis.
𝑓 𝑥 = 𝑥 3 + 𝑥 2 − 2𝑥 + 1
−𝑓 𝑥 = − 𝑥 3 + 𝑥 2 − 2𝑥 + 1
= −𝑥 3 − 𝑥 2 + 2𝑥 − 1
112
Graphing with Transformations (reflection)
If you know the graph of a function, it can be easy to graph a similar function using transformations.
Changing the sign of the 𝑥 in the function creates a reflection, or mirror image, of the graph across
the 𝑦-axis.
𝑓 −𝑥 = −𝑥 3 + −𝑥 2 − 2 −𝑥 + 1
𝑓 𝑥 = 𝑥 3 + 𝑥 2 − 2𝑥 + 1
= −𝑥 3 + 𝑥 2 + 2𝑥 − 1
113
Example - Graphing with Transformations
Given the function 𝑓 𝑥 = 𝑥 4 , find the function that is the result of an
downward shift of 3, a shift left of 6, and a reflection about the 𝑥-axis.
114
Example - Graphing with Transformations
The function 𝑝 = 𝑟 𝑧 = − 𝑧 − 2 + 3 is obtained from a linear transformation of 𝑡 𝑧 = 𝑧.
Determine the transformations imposed on 𝑡 to get the graph of 𝑝.
115
Increasing, Decreasing, Constant
If a graph is increasing, then it
goes up as you move from left
to right.
If a graph is decreasing, then it
goes down as you move from
left to right.
116
Increasing, Decreasing, Constant (cont’d)
If a graph is constant, then it doesn’t increase or decrease as you move from left
to right.
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Example - Increasing, Decreasing
When asked for the interval, look for the 𝑥 values.
The function is increasing on the interval:
The function is decreasing on the interval:
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Example - Increasing, Decreasing
The function is increasing on the interval:
The function is decreasing on the interval:
The function is constant on the interval:
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Absolute Extrema
The absolute maximum is the highest point on the
graph.
The absolute minimum is the lowest point on the
graph.
The extrema might not be at the end points.
Absolute max:
Absolute min:
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Relative Extrema
A relative maximum is a point on the graph that is
higher than the points around it.
The relative minimum is a point on the graph that is
lower than the points around it.
This function has a relative maximum of _____ at
𝑥 = _____
This function has a relative minimum of _____ at
𝑥 = _____
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Example - Application
A truck rental company rents a moving truck for one day by charging $39 and
$0.60 per mile.
1. Write a linear equation that relates the cost C, in dollars, of renting the truck
to the number x of miles driven.
2. What is the cost of renting the truck if the truck is driven 110 miles?
3. How far could the truck be driven if the cost must be less than $150?
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Example - Application
A company is producing very large stuffed animals. The variable cost is $7 per stuffed animal and
the fixed costs are $25,000. They will sell the stuffed animals for $76 each. Let 𝑥 be the number
of stuffed animals produced.
a)
Write the total cost 𝐶 as a function of the number of stuffed animals produced.
b) Write the total revenue 𝑅 as a function of the number of stuffed animals produced.
c)
Write the total profit 𝑃 as a function of the number of stuffed animals produced.
d) Find the number of stuffed animals which must be produced to break even.
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