Sets
A set is a collection of objects.
A well-defined set has no ambiguity as to what objects are in the
set or not.
For example:
The collection of all red cars
The collection of positive numbers
The collection of people born before 1980
The collection of greatest baseball players
All of these collections are sets. However, the collection of greatest
baseball players is not well-defined.
Normally we restrict our attention to just well-defined sets.
Functions
A function consists of 3 things:
1. A set called the domain
2. A set called the range
3. A rule that gives a relationship between the domain and range
We write f : A → B to mean that the function f has domain A
and range B.
A function can be thought of as taking something from the domain
and returning something in the range.
In 2120 our functions will normally “map” numbers to numbers
(i.e. the domain and range of our functions are sets of numbers).
Thus, we often will not explicitly define the domain and range of
our function.
However, there are other types of functions. For example, the
function that maps final averages to letter grades.
Functions
A function has a rule that gives a relationship between the domain
and range. Given an item from the domain this rule must return
exactly one value from the range.
This rule is often given as an equation.
For example, f (x) = 2x + 1. This rule says to map an object to
two times itself plus 1.
f (0)
f (1)
f (5)
f (3.5)
f (y)
f (z + 2)
=
=
=
=
=
=
Functions
A function has a rule that gives a relationship between the domain
and range. Given an item from the domain this rule must return
exactly one value from the range.
This rule is often given as an equation.
For example, f (x) = 2x + 1. This rule says to map an object to
two times itself plus 1.
f (0)
f (1)
f (5)
f (3.5)
f (y)
f (z + 2)
=
=
=
=
=
=
2·0+1=1
Functions
A function has a rule that gives a relationship between the domain
and range. Given an item from the domain this rule must return
exactly one value from the range.
This rule is often given as an equation.
For example, f (x) = 2x + 1. This rule says to map an object to
two times itself plus 1.
f (0)
f (1)
f (5)
f (3.5)
f (y)
f (z + 2)
=
=
=
=
=
=
2·0+1=1
2·1+1=3
Functions
A function has a rule that gives a relationship between the domain
and range. Given an item from the domain this rule must return
exactly one value from the range.
This rule is often given as an equation.
For example, f (x) = 2x + 1. This rule says to map an object to
two times itself plus 1.
f (0)
f (1)
f (5)
f (3.5)
f (y)
f (z + 2)
=
=
=
=
=
=
2·0+1=1
2·1+1=3
2 · 5 + 1 = 11
Functions
A function has a rule that gives a relationship between the domain
and range. Given an item from the domain this rule must return
exactly one value from the range.
This rule is often given as an equation.
For example, f (x) = 2x + 1. This rule says to map an object to
two times itself plus 1.
f (0)
f (1)
f (5)
f (3.5)
f (y)
f (z + 2)
=
=
=
=
=
=
2·0+1=1
2·1+1=3
2 · 5 + 1 = 11
2 · 3.5 + 1 = 8
Functions
A function has a rule that gives a relationship between the domain
and range. Given an item from the domain this rule must return
exactly one value from the range.
This rule is often given as an equation.
For example, f (x) = 2x + 1. This rule says to map an object to
two times itself plus 1.
f (0)
f (1)
f (5)
f (3.5)
f (y)
f (z + 2)
=
=
=
=
=
=
2·0+1=1
2·1+1=3
2 · 5 + 1 = 11
2 · 3.5 + 1 = 8
2y + 1
Functions
A function has a rule that gives a relationship between the domain
and range. Given an item from the domain this rule must return
exactly one value from the range.
This rule is often given as an equation.
For example, f (x) = 2x + 1. This rule says to map an object to
two times itself plus 1.
f (0)
f (1)
f (5)
f (3.5)
f (y)
f (z + 2)
=
=
=
=
=
=
2·0+1=1
2·1+1=3
2 · 5 + 1 = 11
2 · 3.5 + 1 = 8
2y + 1
2(z + 2) + 1 = 2z + 5
Graphs
We often think of a graph as a “picture” of a function.
The graph of a function f is the set of all points (x, y) that satisfy
the equation y = f (x)
Consider the function f (x) = x2 + 1. The graph of f is all of the
points that satisfy y = x2 + 1.
x
y
0
1
1
2
-1
2
2
5
-2
5
3
10
-3
10
Graphs
We often think of a graph as a “picture” of a function.
The graph of a function f is the set of all points (x, y) that satisfy
the equation y = f (x)
Consider the function f (x) = x2 + 1. The graph of f is all of the
points that satisfy y = x2 + 1.
x
y
0
1
1
2
-1
2
2
5
-2
5
3
10
-3
10
Graphs
We often think of a graph as a “picture” of a function.
The graph of a function f is the set of all points (x, y) that satisfy
the equation y = f (x)
Consider the function f (x) = x2 + 1. The graph of f is all of the
points that satisfy y = x2 + 1.
x
y
0
1
1
2
-1
2
2
5
-2
5
3
10
In this case we “know” the
graph is a parabola so only
a few points are needed.
-3
10
Lines
Any function that can be put into the following form is called a
linear function.
f (x) = mx + b
This form is called Slope-Intercept Form:
m is the slope of the line and tells how steep the line is
b is the y-component of the y-intercept (0, b). The point
(0, b) always satisfies y = mx + b and is where the line “hits”
the y-axis.
Graphing lines takes just two points.
Lines
Consider the linear function f (x) = −2x + 1.
The slope is
The y-intercept is
Graph f (x) = −2x + 1
Lines
Consider the linear function f (x) = −2x + 1.
The slope is −2
The y-intercept is
Graph f (x) = −2x + 1
Lines
Consider the linear function f (x) = −2x + 1.
The slope is −2
The y-intercept is (0, 1)
Graph f (x) = −2x + 1
Lines
Consider the linear function f (x) = −2x + 1.
The slope is −2
The y-intercept is (0, 1)
Graph f (x) = −2x + 1
Slope
Slope Formula: Suppose you know two distinct points on a line,
(x1 , y1 ) and (x2 , y2 ). If x1 6= x2 then
m=
change in y
“rise”
y2 − y1
=
=
.
x2 − x1
change in x
“run”
If x1 = x2 , then m is undefined and the line is vertical.
What is the slope of the line through the points (1, 2) and (3, 6)?
Slope
Slope Formula: Suppose you know two distinct points on a line,
(x1 , y1 ) and (x2 , y2 ). If x1 6= x2 then
m=
change in y
“rise”
y2 − y1
=
=
.
x2 − x1
change in x
“run”
If x1 = x2 , then m is undefined and the line is vertical.
What is the slope of the line through the points (1, 2) and (3, 6)?
m=
6−2
4
= =2
3−1
2
Slope
When you know the slope of a line and a point on the line it is
easy to find more points on the line. Just move “over” by the run
and “up” by the rise.
f (x) = −2x + 1, m = − 21 , y-intercept = (0, 1)
Slope
When you know the slope of a line and a point on the line it is
easy to find more points on the line. Just move “over” by the run
and “up” by the rise.
f (x) = −2x + 1, m = − 21 , y-intercept = (0, 1)
Slope
When you know the slope of a line and a point on the line it is
easy to find more points on the line. Just move “over” by the run
and “up” by the rise.
f (x) = −2x + 1, m = − 21 , y-intercept = (0, 1)
Slope
When you know the slope of a line and a point on the line it is
easy to find more points on the line. Just move “over” by the run
and “up” by the rise.
f (x) = −2x + 1, m = − 21 , y-intercept = (0, 1)
Slope
When you know the slope of a line and a point on the line it is
easy to find more points on the line. Just move “over” by the run
and “up” by the rise.
f (x) = −2x + 1, m = − 21 , y-intercept = (0, 1)
Slope
Slope also gives us a general feel for how the line looks even
without graphing it.
Positive slope
Negative slope
Zero slope
Undefined slope
Point-Slope Form
While slope-intercept form requires both the slope and the
y-intercept if we know the slope and any point on the line we can
find an equation for the line.
Point-Slope Form: Given a point (x1 , y1 ) on a line and its slope m
an equation for the line is:
y − y1 = m(x − x1 )
How can we turn point-slope form into slope-intercept form?
Point-Slope Form
While slope-intercept form requires both the slope and the
y-intercept if we know the slope and any point on the line we can
find an equation for the line.
Point-Slope Form: Given a point (x1 , y1 ) on a line and its slope m
an equation for the line is:
y − y1 = m(x − x1 )
How can we turn point-slope form into slope-intercept form?
We can transform point-slope form into slope-intercept form by
solving for y.
y = mx − mx1 + y1 , b = y1 − mx1
Point-Slope Form
Give an equation in point-slope form for the line with slope m = 2
that passes through the point (1, 2).
What is the y-intercept?
Point-Slope Form
Give an equation in point-slope form for the line with slope m = 2
that passes through the point (1, 2).
y − 2 = 2(x − 1)
What is the y-intercept?
Point-Slope Form
Give an equation in point-slope form for the line with slope m = 2
that passes through the point (1, 2).
y − 2 = 2(x − 1)
What is the y-intercept?
y = 2x − 2 + 2
= 2x
So b = 0 and the y-intercept is (0, 0).
Two Point Form
Suppose we are given two distinct points on a line. How can we
find an equation for the line?
Two Point Form
Suppose we are given two distinct points on a line. How can we
find an equation for the line?
Find the slope and use one of the points to write a point-slope
form equation.
y − y1 =
y2 − y1
(x − x1 )
x2 − x1
Parallel and Perpendicular Lines
Parallel lines are either equal (the same line) or never touch. In
either case they have the same slope (or both slopes are
undefined).
Perpendicular lines meet at 90◦ . If two lines are perpendicular and
m1 is the slope of one line and m2 is the slope of the other, then:
m1 m2 = −1,
m2 = − m11 ,
m1 = 0, m2 = ∞,
or m1 = ∞, m2 = 0.
Cost Function
Linear functions are important because they are simple but
expressive.
Many real world processes can be modeled with linear functions.
Cost function:
C(x) = ax + b
where
x is the number of items
b is the fixed cost (in dollars)
a is the unit cost (in dollars)
C(x) is the total cost to make x items (in dollars)
Costs
The fixed cost is the amount a manufacturer has to pay regardless
of how many items are produced (e.g. rent, utilities, setup,...)
The unit cost is the amount a manufacturer has to pay per item
(e.g. supplies, ingredients,...)
Cost Function Example
Suppose a band wants to make some CDs of their work to sell.
The CD burner cost $50 and each blank CD cost $0.25.
What is their cost function?
How much would it cost to make 200 CDs?
How many CDs can they make with $75?
Cost Function Example
Suppose a band wants to make some CDs of their work to sell.
The CD burner cost $50 and each blank CD cost $0.25.
What is their cost function?
C(x) = 0.25x + 50
How much would it cost to make 200 CDs?
How many CDs can they make with $75?
Cost Function Example
Suppose a band wants to make some CDs of their work to sell.
The CD burner cost $50 and each blank CD cost $0.25.
What is their cost function?
C(x) = 0.25x + 50
How much would it cost to make 200 CDs?
C(200) = 0.25 · 200 + 50
= 50 + 50
= 100
How many CDs can they make with $75?
Cost Function Example
Suppose a band wants to make some CDs of their work to sell.
The CD burner cost $50 and each blank CD cost $0.25.
What is their cost function?
C(x) = 0.25x + 50
How much would it cost to make 200 CDs?
C(200) = 0.25 · 200 + 50
= 50 + 50
= 100
How many CDs can they make with $75?
C(x) = 75
0.25x + 50 = 75
0.25x = 25
x = 100
Unknown Costs
Many times a business can not separate their fixed costs and unit
costs.
However, they can be determined if they know their total costs for
producing two different amounts of product.
Suppose a shop-owner knows they can make 10 suits in a week for
$360 and 7 suits in a week for $270.
How can we find the cost function?
What are the two points?
Unknown Costs
Many times a business can not separate their fixed costs and unit
costs.
However, they can be determined if they know their total costs for
producing two different amounts of product.
Suppose a shop-owner knows they can make 10 suits in a week for
$360 and 7 suits in a week for $270.
How can we find the cost function?
Determine the cost function by finding the slope.
What are the two points?
Unknown Costs
Many times a business can not separate their fixed costs and unit
costs.
However, they can be determined if they know their total costs for
producing two different amounts of product.
Suppose a shop-owner knows they can make 10 suits in a week for
$360 and 7 suits in a week for $270.
How can we find the cost function?
Determine the cost function by finding the slope.
What are the two points?
(10, 360), (7, 270)
So the slope is m =
360−270
10−7
=
90
3
= 30.
Unknown Costs
Now that we know the slope and a point we can find the a
point-slope form equation for the line.
y − 360 = 30(x − 10)
We then solve for y to find the cost function.
y = 30(x − 10) + 360
= 30x − 300 + 360
= 30x + 60
C(x) = 30x + 60
Their fixed cost is
Their unit cost is
.
.
Unknown Costs
Now that we know the slope and a point we can find the a
point-slope form equation for the line.
y − 360 = 30(x − 10)
We then solve for y to find the cost function.
y = 30(x − 10) + 360
= 30x − 300 + 360
= 30x + 60
C(x) = 30x + 60
Their fixed cost is $60.
Their unit cost is
.
Unknown Costs
Now that we know the slope and a point we can find the a
point-slope form equation for the line.
y − 360 = 30(x − 10)
We then solve for y to find the cost function.
y = 30(x − 10) + 360
= 30x − 300 + 360
= 30x + 60
C(x) = 30x + 60
Their fixed cost is $60.
Their unit cost is $30.
Revenue Function
Revenue function:
R(x) = sx
where
x is the number of items sold
s is the sell price of the item
R(x) is the total revenue from selling x items
Supposed the shop owner sell their suits for $50 each.
R(x) = 50x
Break-Even Point
The break-even point is the point where total cost equals total
revenue.
R(x) = C(x)
What is the break-even point for our suit selling shop owner
(remember the break-even point is a point)?
C(x) = 30x + 60
R(x) = 50x
Break-Even Point
The break-even point is the point where total cost equals total
revenue.
R(x) = C(x)
What is the break-even point for our suit selling shop owner
(remember the break-even point is a point)?
C(x) = 30x + 60
R(x) = 50x
50x = 30x + 60
20x = 60
x=3
Break-Even Point
The break-even point is the point where total cost equals total
revenue.
R(x) = C(x)
What is the break-even point for our suit selling shop owner
(remember the break-even point is a point)?
C(x) = 30x + 60
R(x) = 50x
50x = 30x + 60
20x = 60
x=3
Now that we know x we need to solve for y.
y = 50 · 3
= 150
The break even point is: (3, 150)
Break-Even Point
We can solve for the break-even point algebraically.
R(x) = C(x)
sx = ax + b
(s − a)x = b
b
x = s−a
Thus,
sb
b
s−a , s−a
is the general break-even point.
For our suit selling shopowner, a = 30,
b = 60, and s = 50. So
50·60
60
the break-even point is 50−30 , 50−30 = (3, 150).
However, it is “better” to remember the process of setting total
cost equal to total revenue than to memorize the break-even point
formula.
Break-Even Point
Geometrically the break-even point occurs where the graphs of
C(x) and R(x) intersect.