1. Relations & Functions
A relation is a set of ordered ( 𝑥 , 𝑦 ) pairs. Sometimes they are ( input , output ) values such as ( age , height )

2. Relations & Functions
A relation is a set of ordered ( 𝑥 , 𝑦 ) pairs. Sometimes they are ( input , output ) values such as ( age , height )
This set { 15 , 65 , 21 , 70 , , 73 , 16 , 62 } is a relation.

3. Relations & Functions
A relation is a set of ordered ( 𝑥 , 𝑦 ) pairs. Sometimes they are ( input , output ) values such as ( age , height )
This set { 15 , 65 , 21 , 70 , , 73 , 16 , 62 } is a relation.
The domain of a relation are the input or x – values { 15 , 21 , 18 , 16 }

4. Relations & Functions
A relation is a set of ordered ( 𝑥 , 𝑦 ) pairs. Sometimes they are ( input , output ) values such as ( age , height )
This set { 15 , 65 , 21 , 70 , , 73 , 16 , 62 } is a relation.
The domain of a relation are the input or x – values { 15 , 21 , 18 , 16 }
The range of a relation are the output or y – values { 65 , 70 , 73 , 62 }

5. Relations & Functions ** ALL SETS OF ORDERED PAIRS ARE RELATIONS
A relation is a set of ordered ( 𝑥 , 𝑦 ) pairs. Sometimes they are ( input , output ) values such as ( age , height )
This set { 15 , 65 , 21 , 70 , , 73 , 16 , 62 } is a relation.
The domain of a relation are the input or x – values { 15 , 21 , 18 , 16 }
The range of a relation are the output or y – values { 65 , 70 , 73 , 62 }
** ALL SETS OF ORDERED PAIRS ARE RELATIONS

6. Relations & Functions
A function is a set of ordered ( 𝑥 , 𝑦 ) pairs or ( inputs , outputs ) where the domain values are all unique. In other words, x – values or input values can not repeat or be represented more than once.

7. Relations & Functions
A function is a set of ordered ( 𝑥 , 𝑦 ) pairs or ( inputs , outputs ) where the domain values are all unique. In other words, x – values or input values can not repeat or be represented more than once.
This set , 2 , 1 , 3 , 2 , 4 , 3 , is a function. No x – values repeat.

8. Relations & Functions
A function is a set of ordered ( 𝑥 , 𝑦 ) pairs or ( inputs , outputs ) where the domain values are all unique. In other words, x – values or input values can not repeat or be represented more than once.
This set , 2 , 1 , 3 , 2 , 4 , 3 , is a function. No x – values repeat.
This set (1, 5 , −1 , 1 , 0 , 0 , −1 , 7 , 2 , 4 } IS NOT a function because
x – value of – 1 repeats or “shows up” more than once.

9. Relations & Functions
Example : Does the table below represent a function ?
Laps ( x ) 3 8 5 6 4
Minutes ( y ) 9 14 15

10. Relations & Functions
Example : Does the table below represent a function ?
Laps ( x ) 3 8 5 6 4
Minutes ( y ) 9 14 15

11. Relations & Functions
Example : Does the table below represent a function ?
A quick scan of the x – values show none of them repeating so YES, this table is a function.
Laps ( x ) 3 8 5 6 4
Minutes ( y ) 9 14 15

12. Relations & Functions
A vertical line test can be used to test whether a given graph is a function. If a vertical line drawn thru the graph intersects the graph more than once, the graph DOES NOT represent a function.

13. Relations & Functions
A vertical line test can be used to test whether a given graph is a function. If a vertical line drawn thru the graph intersects the graph more than once, the graph DOES NOT represent a function.
Here is a linear function :
Test if it is a function with the vertical line test…

14. Relations & Functions
A vertical line test can be used to test whether a given graph is a function. If a vertical line drawn thru the graph intersects the graph more than once, the graph DOES NOT represent a function.
Here is a linear function :
Test if it is a function with the vertical line test…
No vertical line we draw crosses the graph more than once so this graph represents a function.

15. Relations & Functions
A vertical line test can be used to test whether a given graph is a function. If a vertical line drawn thru the graph intersects the graph more than once, the graph DOES NOT represent a function.
Does this graph represent a function?

16. Relations & Functions
A vertical line test can be used to test whether a given graph is a function. If a vertical line drawn thru the graph intersects the graph more than once, the graph DOES NOT represent a function.
Does this graph represent a function?
No, we only need to find one line that intersects the graph more than once.

17. Relations & Functions
A vertical line test can be used to test whether a given graph is a function. If a vertical line drawn thru the graph intersects the graph more than once, the graph DOES NOT represent a function.
This function has some opened and closed circles.
The point ( 5 , 4 ) has an open circle which is not part of the graph.
The points ( - 8 , 2 ) and ( 5 , - 2 ) are closed circles and considered part of the graph.

18. Relations & Functions
A vertical line test can be used to test whether a given graph is a function. If a vertical line drawn thru the graph intersects the graph more than once, the graph DOES NOT represent a function.
Even though the vertical line appears to cross more than once, one of the points is open and not part of the graph.
So this graph is a function.

19. Relations & Functions
Mapping functions pairs the domain ( x – values ) with their range ( y – values ) separately in sets.

20. Relations & Functions
Mapping functions pairs the domain ( x – values ) with their range ( y – values ) separately in sets.
EXAMPLE : Map the given function
−2 , 2 , −1 , 5 , 0 , 8 , ( 1 , 11 )

21. Relations & Functions
Mapping functions pairs the domain ( x – values ) with their range ( y – values ) separately in sets.
EXAMPLE : Map the given function
−2 , 2 , −1 , 5 , 0 , 8 , ( 1 , 11 )
DOMAIN
RANGE −2 −1 1 2 5 8 11
This mapping shows a function as all domain values are paired with only one range value…

22. Relations & Functions EXAMPLE : Does this mapping show a function ?
DOMAIN
RANGE 10 −5 3 6 −2 5

23. Relations & Functions EXAMPLE : Does this mapping show a function ?
DOMAIN
RANGE 10 −5 3 6 −2 5
No, one domain value maps into more than one range value…
- 2 shares – 5 and 0

24. Relations & Functions
EXAMPLE # 2 : Does this mapping show a function ?
DOMAIN
RANGE −2 2 4

25. Relations & Functions
EXAMPLE # 2 : Does this mapping show a function ?
DOMAIN
RANGE
Yes, even though -2 and + are pointing to the same range value, each domain value is still paired only one time −2 2 4