Tuesday
Turn Homework into Basket
Parent Function Quiz
• 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
1 2 63 4 5 7 8 9 10
4
3
2
7
56
8
9
x-axis
y-axis
0
1-2-6 -3-4-5-7-8-910
-4
-3
-2
-1
-7
-5
-6
-8
-9
0
-1
Quadrant I
(+, +)
Quadrant II
(-, +)
Quadrant IV
(+, -)
Quadrant III
(-, -)
Origin (0,0)
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
x-axis
DomainInput
Independent Variable
y-axisRange
OutputD
epen
dent
Var
iabl
e
Example 1: Domain/Range
• 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
RelationsOrdered Pairs
{(2, 3),(-3, 1),(1, -2)} X Y
2 3
-3 1
1 -2
Data Tables
GraphsMapping
2
-3
1
3
1
-2
X Y
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
Tell whether the pairing is a function.
Identify a functionEXAMPLE 2
a.
NOT a function because the input “0” is paired with both 2 and 3.
b.
Identify a functionEXAMPLE 2
OutputInput
21
0 0
4 8
6 12
Function? Yes
GUIDED PRACTICE for Example 2
Tell whether the pairing is a function.
1221Output
12963Input2.
Function? Yes
GUIDED PRACTICE for Example 2
Tell whether the pairing is a function.
3210Output
7422Input3.
Function? No
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
y 2 = x
x
y
Vertical Line Test
Consider the graphs.
x 2 + y
2 = 1
x
y
y = x 2
x
y
2 points of intersection
1 point of intersection
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
x
y
4
-4
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.
Domain
Range
x
y
– 1
1
Example: Find the domain and range of the function f (x) = from its graph.
The domain is [–3,∞).
The range is [0,∞).
3x
Range
Domain
(–3, 0)
These are "assumed" arrows!
The graph goes on forever.
Example 1Domain( , )
Range[ 3, )
Example 2
Domain( , )
Range( , 4]
Example 3
Domain[0, )
Range( , )
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
Domain( , )
Range[2, )
6
4
2
-2
-4
-6
-5 5
Domain( ,3]
Range[1, )
Domain( , )
Range
[0, )
Domain( , 1) [1,6]U
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