Download Relations and Functions

Thursday
Thursday
Turn Homework into Basket
1
Get sheet of paper
2
Solve and graph
3(  x  3)  7  ( x  8)
Write answer Interval Notation
in your notebooks
INQ : x  4
INT : (4, )
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.
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 { } 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
•
•
•
•
•
•
•
•
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}
GUIDED PRACTICE
1.
for Example 1
Identify the domain and range of the function.
Input
Output
0
5
1
2
2
2
4
1
SOLUTION
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
Relations
Ordered Pairs
(2, 3)
(-3, 1)
(1, -2)
Graphs
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. 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.
EXAMPLE 2
Identify a function
Tell whether the pairing is a function.
a.
The pairing is 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
The pairing is a function because each input is paired
with exactly one output.
for Example 2
GUIDED PRACTICE
Tell whether the pairing is a function.
2.
Input
Output
SOLUTION
3
1
3
6
9
12
6
2
9
2
12
1
1
2
2
1
The pairing is a function because each input is paired
with exactly one output.
for Example 2
GUIDED PRACTICE
Tell whether the pairing is a function.
3.
Input
Output
SOLUTION
2
0
2
4
7
2
1
4
2
7
3
0
1
2
3
The pairing is not a function because each input is not
paired with exactly one output.
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
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
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