functions 3.3 is a function because each first element is paired with a different second element....
TRANSCRIPT
![Page 1: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/1.jpg)
Functions3.3
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is a function because each first element is paired with a different second element.
The relation
(2,3), (5,9), (1,0), (10, 2)A
(4,5), (4,0), ( 1,9)B
The relation
is NOT a function because (4, 5) and (4, 0) have the same first element and different second elements.
Which of the following are functions?
(9,0), (3,8), (5,8), (9, 1)
(0, ), (9, ), ( 2, ), ( 5, )
( 3,5), (7, 2), ( 3,2), (9, 1)
(0,1), (1,0), (1,1)
A
B t e q b
C
D
No
Yes
No
No
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Vertical Line Test
There is a graphical method for deciding if a relation is a function. It’s called the vertical line
test. If a graph touches a vertical line more than once, at any place on the graph,
it is not a function.
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Vertical Line Test – Does a vertical line touch two points of the graphed relation at once?
If so, it is not a function.
![Page 5: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/5.jpg)
Example 6-1a
Determine whether the relation is a function. Explain.
Yes, it’s a function.
![Page 6: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/6.jpg)
Determine whether the relation is a function. Explain.
x y–7 –12
–4 –9
2 –3
5 0
Yes, it’s a function.
![Page 7: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/7.jpg)
Example 6-1c
Determine whether {(–5, 2), (–2, 5), (0, 7), (0, 9)} is a function. Explain.
No, it’s not a function.
![Page 8: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/8.jpg)
Determine whether each relation is a function. Explain.
a.
Yes
![Page 9: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/9.jpg)
b.
c. {(3, 0), (1, 2), (4, 0), (5, –1)}
–13
–42
–21
23
YX
No
Yes
![Page 10: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/10.jpg)
Determine whether is a function.
No
![Page 11: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/11.jpg)
Determine whether is a function.
Answer: yes
![Page 12: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/12.jpg)
Determine whether this graph is a function.
Answer: yes
![Page 13: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/13.jpg)
Example 6-2b
Determine whether this graph is a function.
Answer: no
![Page 14: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/14.jpg)
The solid dot shows that (3, 2) belongs to the graph. The open dot shows that (3, 4) does not belong to the graph.
![Page 15: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/15.jpg)
Write these answers in your notes: Does this graph represent a function?
Yes
![Page 16: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/16.jpg)
Does this graph represent a function?
Yes
![Page 17: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/17.jpg)
Does this graph represent a function?
Yes
![Page 18: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/18.jpg)
Does this graph represent a function?
No
![Page 19: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/19.jpg)
FunctionNotation
![Page 20: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/20.jpg)
( , ) (1,4), (2,3), (3,2), (4, 8), (5,2)g x y
( , ) ( 4,0), (9,1), ( 3, 2), (6,6), (0, 2)h x y
Using Function NotationFind g(2) and g(5) for
.
g(2) = ____ g(5) = ____
Consider
What is h(9)? ___ h(6)?____ h(0)? _____
3 2
1 62
![Page 21: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/21.jpg)
Now look away from your notes for a while.
![Page 22: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/22.jpg)
Instead of y in 2 3y x we write f(x).
doesn’t mean f times x. ( )f x
It means the value of the function calledf at the point x.
( ) 2 3f x x
![Page 23: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/23.jpg)
If find .
Multiply.
Subtract.
Answer:
Replace x with 4.
![Page 24: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/24.jpg)
If find .
Multiply.
Subtract.
Answer:
Replace x with –5.
![Page 25: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/25.jpg)
If find .
Distributive Property
Simplify.
Answer:
Replace x with 2 – x.
![Page 26: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/26.jpg)
If find each value.
a.
b.
c.
Answer: 11
Answer: –11
Answer:
Write on your notes
![Page 27: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/27.jpg)
If , find .
Multiply.
Simplify.
Answer:
Replace m with –3.
![Page 28: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/28.jpg)
Answer: 8
If find each value.
Write on your notes
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IfMultiple-Choice Test Item
A 69 . B 70. C 79. D 81.
Read the Test Item
The symbol is just a different notation for f(x). Solve the Test Item
Replace x with –5.
Answer: A
Think: .
Replace x with –5.
Simplify.
![Page 30: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/30.jpg)
Answer: C
SAT Question
A 35. B 30. C 20. D 19.If
![Page 31: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/31.jpg)
Finding the domain of a function from an equation:
What is the domain of f for ?4
( )3
xf x
x
Therefore, . That means that The domain of the functionis
3 0x 3.x
3x x
This implies that x is all real numbers except -3.
0
xAs we know, is undefined.
![Page 32: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/32.jpg)
Try this. What is the domain of each?
( )( 1)( 3)
xg x
x x
( ) 3 9h x x
2 1( )
3 3
xp x
x
1, 3x x
x x
0x x
![Page 33: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/33.jpg)
Classwork: 4, 18, 22/119-120
![Page 34: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/34.jpg)
Get ready for a “Small Quiz” to be written
on your grade sheet.
![Page 35: Functions 3.3 is a function because each first element is paired with a different second element. The relation is NOT a function because (4, 5) and (4,](https://reader036.vdocuments.us/reader036/viewer/2022062322/5697bfaa1a28abf838c9a90d/html5/thumbnails/35.jpg)
The End
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Quiz. Copy the problems and write the answer.
Put your grade paper on the front of your row, quiz side down.
1. Write the Cartesian Product A X B if A = {0, 1, 2} and B = {6, 7}.
2.Write the domain of A if A = {(7, 3), (-2,5), (7,1), (8, 9)}