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Thursday
Thursday
Turn Homework into Basket
1Get sheet of paper
2
Solve and graph
3( 3) 7 ( 8)x x
Write answer Interval Notation
in your notebooks
: 4
: ( 4, )
INQ x
INT
Relations and Functions
Chapter 2
Section 2-1
Pages 72-81
Objectives• I can find Domain and Range• I can determine if the relation is a
function by two methods• IMPORTANT VOCABULARY in
this section!!
Important Vocabulary
• Relation• Domain• Range• Function• Vertical Line Test
Cartesian Coordinate Plot
• x-axis: This is the horizontal axis.• y-axis: This is the vertical axis• Origin: This is the center point (0,0)
• Each point on the coordinate plane can be represented by an ordered pair (x,y), where x is the distance from Origin on the X-Axis and y is the distance from Origin on the Y-Axis.
Quadrants
• The coordinate system is divided into 4 quadrants.
• Look at the overhead slide 2-1.1 to see how this entire system works.
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 { } 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}
SOLUTION
GUIDED PRACTICE for Example 1
1. Identify the domain and range of the function.
Input 0 1 2 4Output 5 2 2 1
The domain is the set of inputs: {0,1,2,4}
The range is the set of outputs: {1,2,5}
4 Ways to see Relations
RelationsOrdered Pairs
(2, 3)
(-3, 1)
(1, -2)
X Y
2 3
-3 1
1 -2
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. Each element from the domain
has only 1 element from the range
Ex 2: How about this relation. Is it a function?
• Given the following { (2,3), (-4,8), (2,6), (7,-3)}• Domain?• { -4, 2, 7}• Range?• {-3, 3, 6, 8}• Function: No. the element 2 from the domain is
mapped to 2 elements in the range.
Tell whether the pairing is a function.
Identify a functionEXAMPLE 2
a.
The pairing is 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
The pairing is a function because each input is pairedwith exactly one output.
SOLUTION
GUIDED PRACTICE for Example 2
Tell whether the pairing is a function.
1221Output
12963Input2.
369
12
1221
The pairing is a function because each input is pairedwith exactly one output.
SOLUTION
GUIDED PRACTICE for Example 2
Tell whether the pairing is a function.
3210Output
7422Input3.
The pairing is not a function because each input is notpaired with exactly one output.
2
47
0123
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 one line intersects the graph at only one point, 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
Examples VLT
VLT Applies to Grid
½ Sheet Activity
• On your ½ sheet of paper• Make 4 relations that we discussed today.• ONE of each type• Trade with your partner and detrmine
– Domain and Range– Is each a function?
Homework
• WS 1-5