Tuesday
Turn Homework into Basket
Parent Function Quiz
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4 Graphs
No Calculators/No Notes
Use entire 10x10 grid
Don’t forget to graph asymptotes if needed!
About 8-10 minutes to complete
Homework Questions?
Relations and
Functions
Chapter 2
Section 2-1
Pages 72-81
Objectives
• I can determine if the relation is a
function by two methods.
• I can find Domain and Range from
relations and continuous graphs
• IMPORTANT VOCABULARY in
this section!!
Important Vocabulary
• Relation
• Domain
• Range
• Discrete Function
• Continuous Function
• Vertical Line Test
yaxis
(-, +)
1
0 -9 -8 -7 -6 -5 -4 -3 -2 -1
Quadrant III
(-, -)
Quadrant I
(+, +)
4
3
2
1
Origin (0,0)
0
Quadrant II
9
8
7
6
5
0 -1 1 2 3 4 5 6 7 8 9 1
0
-2
-3
-4
Quadrant IV
-5
-6
-7
-8
-9
(+, -)
xaxis
Relation
• A relation is a set of ordered
pairs!
• Need the braces { } to show
a set
• Example: { (1, 2), (3, 4), (5, 6) }
Domain and Range
• The domain in any relation is the first coordinates
from the ordered pairs. It is the Input!
• Domain = X -Values
• The range in any relation is the second
coordinates from the ordered pairs. It is the
Output!
• Range = Y- Values
Dependent Variable
y-axis
Range
Output
Independent Variable
x-axis
Domain
Input
Example 1: Domain/Range
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Given the following relation
{(2,3), (-4,8), (2,6), (7,-3)}
What is the Domain?
{ -4, 2, 7}
**Notice they are listed least to greatest!!
No duplicates!!!
What is the Range?
{-3, 3, 6, 8}
Example 2:
• Given the following ordered pairs, find
the domain and range.
• {(4,5), (-2,3), (5,6)}
• Domain is {-2, 4, 5}
• Range is {3, 5, 6}
Answer Format
• When listing a set of numbers for domain or range,
use the set symbols {}
• List numbers from least to greatest (increasing
order). No duplicates!
• Ex: the domain has numbers: 3, -2, 5, 2, 3
• {-2, 2, 3, 5}
4 Ways to see Relations
Relations
Ordered Pairs
{(2, 3),(-3, 1),(1, -2)}
Graphs
Data Tables
X
2
-3
1
Y
3
1
-2
X
Mapping
Y
2
3
-3
1
1
-2
Function
• A function is a special relation in which
• NO DUPLICATED “x-values”
• Example: Is the following relation a function:
{ (1,3), (4,-9), (6,3) }
• Answer: Yes. No x-values are duplicated
Ex 2: How about this relation. Is
it a function?
• Given the following { (2,3), (-4,8), (2,6), (7,-3)}
• Function: No.
• The x-value “2” is duplicated
EXAMPLE 2
Identify a function
Tell whether the pairing is a function.
a.
NOT a function because the input “0” is
paired with both 2 and 3.
EXAMPLE 2
Identify a function
b.
Input
Output
0
0
1
2
4
8
6
12
Function? Yes
GUIDED PRACTICE
for Example 2
Tell whether the pairing is a function.
2.
Input
Output
Function? Yes
3
1
6
2
9
2
12
1
GUIDED PRACTICE
for Example 2
Tell whether the pairing is a function.
3.
Input
Output
Function? No
2
0
2
1
4
2
7
3
Vertical Line Test
• You can use a vertical line test to easily see
on a graph is the relation is a function.
• You place a straight edge like a pencil
vertical on the graph and move it across the
graph. If the line intersects the graph at
only one point at a time, then it is a
function.
Applying VLT
Vertical Line Test
Consider the graphs.
y
y
y
y2 = x
y = x2
x
x
2 points
of intersection
1 point
of intersection
x2 + y2 = 1
x
2 points
of intersection
Discrete Function
• A function with ordered
pairs that are just points
and not connected.
Discrete Function
Continuous Functions??
• A function is continuous if it has an infinite
domain and forms a smooth line or curve
• Simply put: It has NO BREAKS!!!
• You should be able to trace it with your
pencil from left to right without picking up
your pencil
27
The domain of a continuous function is all x-values!
We read domain from LEFT to RIGHT
The range of a continuous function is all the y-values!
We read range from BOTTOM to TOP.
y
Range
x
4
-4
Domain
Example: Find the domain and range of the
function f (x) =
x  3 from its graph.
y
Range (–3, 0)
1
–1
Domain
The domain is [–3,∞).
The range is [0,∞).
x
These are "assumed" arrows!
The graph goes on forever.
Example 1
Domain
(, )
Range
[3, )
Example 2
Domain
(, )
Range
(, 4]
Example 3
Domain
[0, )
Range
(, )
8
Domain
6
(, )
4
Range
2
[2, )
-5
5
-2
Domain
6
4
(,3]
Range
[1, )
2
-5
5
Domain
Range
(, )
[0, )
Domain
(, 1) U [1, 6]
Range
(, 6)
What Graph Activity
• Graphs A – Q around the room.
• Answer questions based on domain and
range.
Homework
• WS 1-4
• Quiz Next Class