Math 71A
2.1 and 2.2 – Functions
1
Relations
The number of Drug Law Violations per year at Mt. SAC is given
by the following table:
Year
2007
2008
2009
2010
# of Drug
Law Violations
3
4
2
4
(Source: http://www.mtsac.edu/safety/stats/)
2
Relations
The number of Drug Law Violations per year at Mt. SAC is given
by the following table:
Year
2007
2008
2009
2010
# of Drug
Law Violations
3
4
2
4
(Source: http://www.mtsac.edu/safety/stats/)
We can write this as a set of ordered pairs (called a ______________):
{(2007, 3), (2008, 4), (2009, 2), (2010, 4)}
3
Relations
The number of Drug Law Violations per year at Mt. SAC is given
by the following table:
Year
2007
2008
2009
2010
# of Drug
Law Violations
3
4
2
4
(Source: http://www.mtsac.edu/safety/stats/)
relation
We can write this as a set of ordered pairs (called a ______________):
{(2007, 3), (2008, 4), (2009, 2), (2010, 4)}
4
Relations
{(2007, 3), (2008, 4), (2009, 2), (2010, 4)}
The set of all the first components is called the ______________.
Above, it is _________________.
The set of all the second components is called the ___________.
Above, it is _______________.
5
Relations
{(2007, 3), (2008, 4), (2009, 2), (2010, 4)}
domain
The set of all the first components is called the ______________.
Above, it is _________________.
The set of all the second components is called the ___________.
Above, it is _______________.
6
Relations
{(2007, 3), (2008, 4), (2009, 2), (2010, 4)}
domain
The set of all the first components is called the ______________.
{2007, 2008, 2009, 2010}
Above, it is _________________.
The set of all the second components is called the ___________.
Above, it is _______________.
7
Relations
{(2007, 3), (2008, 4), (2009, 2), (2010, 4)}
domain
The set of all the first components is called the ______________.
{2007, 2008, 2009, 2010}
Above, it is _________________.
range
The set of all the second components is called the ___________.
Above, it is _______________.
8
Relations
{(2007, 3), (2008, 4), (2009, 2), (2010, 4)}
domain
The set of all the first components is called the ______________.
{2007, 2008, 2009, 2010}
Above, it is _________________.
range
The set of all the second components is called the ___________.
{3, 4, 2}
Above, it is _______________.
9
Relations
Relations can be visualized like this (called an “arrow diagram”):
2007
2008
2009
2010
3
2
4
10
Relations
Ex 1.
Find the domain and range of the relation
{(3, 5), (-2, 1), (3,7), (0, 5)}.
11
Relations
Ex 1.
Find the domain and range of the relation
{(3, 5), (-2, 1), (3,7), (0, 5)}.
Domain: {3, -2, 0}
12
Relations
Ex 1.
Find the domain and range of the relation
{(3, 5), (-2, 1), (3,7), (0, 5)}.
Domain: {3, -2, 0}
Range: {5, 1, 7}
13
Functions
A relation in which each member of the domain
corresponds to exactly one member of the range
is called a ____________________.
14
Functions
A relation in which each member of the domain
corresponds to exactly one member of the range
function
is called a ____________________.
15
Functions
A relation in which each member of the domain
corresponds to exactly one member of the range
function
is called a ____________________.
(That is, each input has only one output.)
16
Functions
ex: {(2007, 3), (2008, 4), (2009, 2), (2010, 4)} is a
function since each input has only one output.
ex: {(3, 5), (-2, 1), (3,7), (0, 5)} is not a function since 3
has two outputs: ___ and ___.
17
Functions
ex: {(2007, 3), (2008, 4), (2009, 2), (2010, 4)} is a
function since each input has only one output.
ex: {(3, 5), (-2, 1), (3,7), (0, 5)} is not a function since 3
has two outputs: ___ and ___.
18
Functions
ex: {(2007, 3), (2008, 4), (2009, 2), (2010, 4)} is a
function since each input has only one output.
ex: {(3, 5), (-2, 1), (3,7), (0, 5)} is not a function since 3
5 and ___.
7
has two outputs: ___
19
Functions
ex: {(2007, 3), (2008, 4), (2009, 2), (2010, 4)} is a
function since each input has only one output.
ex: {(3, 5), (-2, 1), (3,7), (0, 5)} is not a function since 3
5 and ___.
7
has two outputs: ___
20
Functions
ex: {(2007, 3), (2008, 4), (2009, 2), (2010, 4)} is a
function since each input has only one output.
ex: {(3, 5), (-2, 1), (3,7), (0, 5)} is not a function since 3
5 and ___.
7
has two outputs: ___
21
Functions as Equations
For this class, we’ll focus on functions in the
form of equations.
For example, 𝑦 = 𝑥 2 − 3𝑥 + 2
22
Functions as Equations
We often give functions names (like 𝑓, 𝑔, ℎ, 𝐹, 𝐺, 𝐻) and
use special notation to define them.
23
Functions as Equations
For example, 𝑓 𝑥 = 𝑥 2 − 3𝑥 + 2.
𝑓 𝑥 is read “𝑓 of 𝑥”.
𝑓(𝑥) represents the value of the function at 𝑥 (that is,
the output of 𝑓).
For example, 𝑓 2 = 2 2 − 3 2 + 2 = 0, that is “𝑓 of
2 is 0”. So, if you input 2, the output is 0.
24
Functions as Equations
For example, 𝑓 𝑥 = 𝑥 2 − 3𝑥 + 2.
𝑓 𝑥 is read “𝑓 of 𝑥”.
𝑓(𝑥) represents the value of the function at 𝑥 (that is,
the output of 𝑓).
For example, 𝑓 2 = 2 2 − 3 2 + 2 = 0, that is “𝑓 of
2 is 0”. So, if you input 2, the output is 0.
25
Functions as Equations
For example, 𝑓 𝑥 = 𝑥 2 − 3𝑥 + 2.
𝑓 𝑥 is read “𝑓 of 𝑥”.
𝑓(𝑥) represents the value of the function at 𝑥 (that is,
the output of 𝑓).
For example, 𝑓 2 = 2 2 − 3 2 + 2 = 0, that is “𝑓 of
2 is 0”. So, if you input 2, the output is 0.
26
Functions as Equations
For example, 𝑓 𝑥 = 𝑥 2 − 3𝑥 + 2.
𝑓 𝑥 is read “𝑓 of 𝑥”.
𝑓(𝑥) represents the value of the function at 𝑥 (that is,
the output of 𝑓).
For example, 𝑓 2 = 2 2 − 3 2 + 2 = 0, that is “𝑓 of
2 is 0”. So, if you input 2, the output is 0.
27
Functions as Equations
Ex 2.
Find ℎ −2 for the function h 𝑥 = 2𝑥 2 + 3𝑥 − 1.
Ex 3.
Find 𝐹 𝑎 + ℎ for 𝐹 𝑥 = 5𝑥 + 7.
Ex 4.
Find 𝑓 71 for 𝑓 𝑥 = 46.
28
Functions as Tables
What are the domain and range?
𝑓 −1 =
𝑥 𝑓 𝑥
-2
4
-1
1
0
0
1
1
2
4
Find 𝑥 such that 𝑓 𝑥 = 4.
29
Functions as Tables
What are the domain and range?
𝑓 −1 =
𝑥 𝑓 𝑥
-2
4
-1
1
0
0
1
1
2
4
Find 𝑥 such that 𝑓 𝑥 = 4.
30
Functions as Tables
What are the domain and range?
Domain: {-2, -1, 0, 1, 2}
Range: {4, 1, 0}
𝑓 −1 =
𝑥 𝑓 𝑥
-2
4
-1
1
0
0
1
1
2
4
Find 𝑥 such that 𝑓 𝑥 = 4.
31
Functions as Tables
What are the domain and range?
Domain: {-2, -1, 0, 1, 2}
Range: {4, 1, 0}
𝑓 −1 =
𝑥 𝑓 𝑥
-2
4
-1
1
0
0
1
1
2
4
Find 𝑥 such that 𝑓 𝑥 = 4.
32
Functions as Tables
What are the domain and range?
Domain: {-2, -1, 0, 1, 2}
Range: {4, 1, 0}
𝑓 −1 = 𝟏
𝑥 𝑓 𝑥
-2
4
-1
1
0
0
1
1
2
4
Find 𝑥 such that 𝑓 𝑥 = 4.
33
Functions as Tables
What are the domain and range?
Domain: {-2, -1, 0, 1, 2}
Range: {4, 1, 0}
𝑓 −1 = 𝟏
𝑥 𝑓 𝑥
-2
4
-1
1
0
0
1
1
2
4
Find 𝑥 such that 𝑓 𝑥 = 4.
34
Functions as Tables
What are the domain and range?
Domain: {-2, -1, 0, 1, 2}
Range: {4, 1, 0}
𝑓 −1 = 𝟏
𝑥 𝑓 𝑥
-2
4
-1
1
0
0
1
1
2
4
Find 𝑥 such that 𝑓 𝑥 = 4.
𝒙 = −𝟐 or 𝒙 = 𝟐
35
Interval Notation
Suppose you wanted to write “the set of all real
numbers between 3 and 5, including 3, but not 5.”
That is, the set of all real numbers 𝑥, such that 3 ≤ 𝑥 <
5.
Here’s how to write it using interval notation:
___________________
36
Interval Notation
Suppose you wanted to write “the set of all real
numbers between 3 and 5, including 3, but not 5.”
That is, the set of all real numbers 𝑥, such that 3 ≤ 𝑥 <
5.
Here’s how to write it using interval notation:
___________________
37
Interval Notation
Suppose you wanted to write “the set of all real
numbers between 3 and 5, including 3, but not 5.”
That is, the set of all real numbers 𝑥, such that 3 ≤ 𝑥 <
5.
Here’s how to write it using interval notation:
___________________
38
Interval Notation
Suppose you wanted to write “the set of all real
numbers between 3 and 5, including 3, but not 5.”
That is, the set of all real numbers 𝑥, such that 3 ≤ 𝑥 <
5.
Here’s how to write it using interval notation:
𝟑, 𝟓
___________________
39
Interval Notation
Inequality
(𝑎, 𝑏)
𝑎<𝑥<𝑏
𝑎, 𝑏
𝑎≤𝑥≤𝑏
𝑎, 𝑏)
𝑎≤𝑥<𝑏
(𝑎, 𝑏
𝑎<𝑥≤𝑏
(𝑎, ∞)
𝑥>𝑎
𝑎, ∞)
𝑥≥𝑎
(−∞, 𝑏)
𝑥<𝑏
(−∞, 𝑏
𝑥≤𝑏
(−∞, ∞)
ℝ
Graph
40
Ex 6.
Interval
Notation
Inequality
Graph
−3 ≤ 𝑥 < 1
−2, 3
𝑥≥2
2.5, 4
(−∞, −1)
41
Ex 6.
Interval
Notation
Inequality
Graph
−3 ≤ 𝑥 < 1
−2, 3
𝑥≥2
2.5, 4
(−∞, −1)
42
Ex 6.
Interval
Notation
Inequality
Graph
−3 ≤ 𝑥 < 1
−2, 3
𝑥≥2
2.5, 4
(−∞, −1)
43
Ex 6.
Interval
Notation
Inequality
Graph
−3 ≤ 𝑥 < 1
−2, 3
𝑥≥2
2.5, 4
(−∞, −1)
44
Ex 6.
Interval
Notation
Inequality
Graph
−3 ≤ 𝑥 < 1
−2, 3
𝑥≥2
2.5, 4
(−∞, −1)
45
Ex 6.
Interval
Notation
Inequality
Graph
−3 ≤ 𝑥 < 1
−2, 3
𝑥≥2
2.5, 4
(−∞, −1)
46
Ex 6.
Interval
Notation
Inequality
Graph
−3 ≤ 𝑥 < 1
−2, 3
𝑥≥2
2.5, 4
(−∞, −1)
47
Ex 6.
Interval
Notation
Inequality
Graph
−3 ≤ 𝑥 < 1
−2, 3
𝑥≥2
2.5, 4
(−∞, −1)
48
Ex 6.
Interval
Notation
Inequality
Graph
−3 ≤ 𝑥 < 1
−2, 3
𝑥≥2
2.5, 4
(−∞, −1)
49
Ex 6.
Interval
Notation
Inequality
Graph
−3 ≤ 𝑥 < 1
−2, 3
𝑥≥2
2.5, 4
(−∞, −1)
50
Ex 6.
Interval
Notation
Inequality
Graph
−3 ≤ 𝑥 < 1
−2, 3
𝑥≥2
2.5, 4
(−∞, −1)
51
If you plot all of a function’s ordered pairs,
then you create the ______ of the function.
To test if a graph is a function we can use the
________________.
If any vertical line intersects a graph in two or
more points, then the graph does not
represent a function.
52
If you plot all of a function’s ordered pairs,
graph of the function.
then you create the ______
To test if a graph is a function we can use the
________________.
If any vertical line intersects a graph in two or
more points, then the graph does not
represent a function.
53
If you plot all of a function’s ordered pairs,
graph of the function.
then you create the ______
To test if a graph is a function we can use the
________________.
If any vertical line intersects a graph in two or
more points, then the graph does not
represent a function.
54
If you plot all of a function’s ordered pairs,
graph of the function.
then you create the ______
To test if a graph is a function we can use the
vertical line test
________________.
If any vertical line intersects a graph in two or
more points, then the graph does not
represent a function.
55
If you plot all of a function’s ordered pairs,
graph of the function.
then you create the ______
To test if a graph is a function we can use the
vertical line test
________________.
If any vertical line intersects a graph in two or
more points, then the graph does not
represent a function.
56
Ex 7.
Which of the following graphs are functions?
57
Ex 7.
Which of the following graphs are functions?
58
Domain and Range of Functions
The ________ of a function is the set of all
possible _________.
The _______ of a function is the set of all
possible _________.
59
Domain and Range of Functions
domain of a function is the set of all
The ________
possible _________.
The _______ of a function is the set of all
possible _________.
60
Domain and Range of Functions
domain of a function is the set of all
The ________
𝒙-values
possible _________.
The _______ of a function is the set of all
possible _________.
61
Domain and Range of Functions
domain of a function is the set of all
The ________
𝒙-values
possible _________.
range of a function is the set of all
The _______
possible _________.
62
Domain and Range of Functions
domain of a function is the set of all
The ________
𝒙-values
possible _________.
range of a function is the set of all
The _______
possible _________.
𝒚-values
63
Domain and Range of Functions
Ex 8.
Use the graph of each function to find its domain and range.
Domain __________________________
Domain __________________________
Range __________________________
Range __________________________
64
Domain and Range of Functions
Ex 8.
Use the graph of each function to find its domain and range.
(−𝟐, 𝟒
Domain __________________________
Domain __________________________
Range __________________________
Range __________________________
65
Domain and Range of Functions
Ex 8.
Use the graph of each function to find its domain and range.
(−𝟐, 𝟒
Domain __________________________
Domain __________________________
(−𝟐, 𝟏
Range __________________________
Range __________________________
66
Domain and Range of Functions
Ex 8.
Use the graph of each function to find its domain and range.
(−𝟐, 𝟒
Domain __________________________
−𝟑, 𝟒)
Domain __________________________
(−𝟐, 𝟏
Range __________________________
Range __________________________
67
Domain and Range of Functions
Ex 8.
Use the graph of each function to find its domain and range.
(−𝟐, 𝟒
Domain __________________________
−𝟑, 𝟒)
Domain __________________________
(−𝟐, 𝟏
Range __________________________
−𝟏, 𝟐
Range __________________________
68
Domain and Range of Functions
Ex 8.
Use the graph of each function to find its domain and range.
Domain __________________________
Domain __________________________
Range __________________________
Range __________________________
69
Domain and Range of Functions
Ex 8.
Use the graph of each function to find its domain and range.
−𝟐, ∞)
Domain __________________________
Domain __________________________
Range __________________________
Range __________________________
70
Domain and Range of Functions
Ex 8.
Use the graph of each function to find its domain and range.
−𝟐, ∞)
Domain __________________________
Domain __________________________
𝟏, ∞)
Range __________________________
Range __________________________
71
Domain and Range of Functions
Ex 8.
Use the graph of each function to find its domain and range.
−𝟐, ∞)
Domain __________________________
−𝟏, 𝟑
Domain __________________________
𝟏, ∞)
Range __________________________
Range __________________________
72
Domain and Range of Functions
Ex 8.
Use the graph of each function to find its domain and range.
−𝟐, ∞)
Domain __________________________
−𝟏, 𝟑
Domain __________________________
𝟏, ∞)
Range __________________________
𝟎, 𝟐
Range __________________________
73
Domain and Range of Functions
Ex 8.
Use the graph of each function to find its domain and range.
Domain __________________________
Range __________________________
74
Domain and Range of Functions
Ex 8.
Use the graph of each function to find its domain and range.
−𝟏, 𝟑)
Domain __________________________
Range __________________________
75
Domain and Range of Functions
Ex 8.
Use the graph of each function to find its domain and range.
−𝟏, 𝟑)
Domain __________________________
{𝟐, 𝟑, 𝟒}
Range __________________________
76
Summary
Here is the same function represented in different ways.
Set of Ordered Pairs
Table
{(1,3), (2, −1), (3,0), (4,3)}
𝑥 𝑓 (𝑥 )
1
3
2 -1
3
0
4
3
Graph
Arrow Diagram
1
2
3
4
3
0
-1
The domain of the above function is _________.
The range of the above function is _________.
77
Summary
Here is the same function represented in different ways.
Set of Ordered Pairs
Table
{(1,3), (2, −1), (3,0), (4,3)}
𝑥 𝑓 (𝑥 )
1
3
2 -1
3
0
4
3
Graph
Arrow Diagram
1
2
3
4
3
0
-1
{𝟏, 𝟐, 𝟑, 𝟒}
The domain of the above function is _________.
The range of the above function is _________.
78
Summary
Here is the same function represented in different ways.
Set of Ordered Pairs
Table
{(1,3), (2, −1), (3,0), (4,3)}
𝑥 𝑓 (𝑥 )
1
3
2 -1
3
0
4
3
Graph
Arrow Diagram
1
2
3
4
3
0
-1
{𝟏, 𝟐, 𝟑, 𝟒}
The domain of the above function is _________.
{𝟑, −𝟏, 𝟎}
The range of the above function is _________.
79