1. Relations and Functions

2. Relation and function function relation
A relation associates the elements of one set with the elements of another set.
A function is a specific type of relation where each element in the domain is associated with exactly one element in the range.
function
relation
*all functions are relations but not all relations are function

3. Function VS Relation
Part 1
Graph

4. Example 1
Explain why the green line is a function while the blue one is only a relation?
Vertical
horizontal

5. One simple method used to determine if a graph is a relation or a function is the vertical line test.
Step 1 :Draw a vertical line at any point on a coordinate plane.
Step 2 :Check whether it intersects the graph once or more .
If it only have one intersection point, it is function, more than it’s a relation.

6. You do
Which one is not a function but a relation?

7. You do
Hyperbola

8. Identify the following chart symbols function or not. Why?

9. You do
Divide them into 2 parts.(Relation or Function)

10. Function VS Relation
Part 2
Arrow Diagram

11. Which one is a function? 1 2 3 4 2 3 4 5 1 2 3 4 2 3 4 5

12. Solution
Keep in mind that once you are determine whether an arrow diagram is a function, the elements in the first column could only relate with one in the second
Easy way: just make sure that in the first column there is just one arrow at each number then it is a function 1 2 3 4 5 1 2 3 4 5

13. You do
Which arrow diagrams represent functions?

14. Function VS Relation
Part 3
Ordered pairs

15. Which one is a function? 1,{ 1,2} {2,3} {3,3} {4,5}
1,{ 1,2} {2,3} {3,3} {4,5}
2,{ 1,2} {1,3} {3,4} {4,5}
Solution
Keep in mind that determining whether an order pair is a function, you just need to make sure that the first element in the order pair is not the same

16. Domain and Range:
The set of first elements of a relation is called the domain. When graphing a relation, the set of first elements will be the x-values of a graph.
The set of related second elements of a relation is called the range. When graphing a relation, the set of second elements will be the y-values of the graph.

17. Independent and dependent
variable

18. For example
The cost for a car rental is 60$, plus 20$ for every 100 km driven
independent
dependent
Distance(km)
Cost($) 60
100 80
200
300
120
400
140
The cost section depends on how much distance it has driven, as a result the distance is the independent variable and the cost is the depend variable

19. In a linear system
The value of the independent variable( the domain) the x value are the independent variable
The value of the dependent variable (the range) the y value are the dependent variable

20. Identify the independent variable and the dependent variable
Identify the independent variable and the dependent variable. Justify your choices. Write the domain and range.

21. Function notation
Function notation: Function notation is a way of showing the independent variable in a function.

22. For example M(n)= 0.8n M of n is equal to 0.8n
This notation shows that M is the dependent variable, and that M depends on n
M(5) represents the value of the function when n=5
M(5) =0.8(5)
M(5) = 4

23. For example P = n -3 T = 5d Y = -5x L = 1/2x + 2
Write in function notation.
C = 20n C(n) = 20n + 8
You try:
P = n -3
T = 5d
Y = -5x
L = 1/2x + 2

24. SKETCHING
A GRAPH

25. example
An oven is turned on at a room temperature of 20*C and it takes 10 minutes to reach a temperature of 190*C A cake is placed in the oven to bake for 30 min The oven is then turned off and returns to room temperature after 15 min The oven is turned on again 45 min later and it takes 15 min to reach a temperature of 160*C Cookies and placed in the oven to bake for 20 min The oven is then turned off and returns to room temperature after 15 min. Sketch the graph

26. 1,TITLE (HEAD) 2,SHAPED(BE FIT) 3,LEGEND(HAND AND FOOT) 4,UNIT (LIMIT)
GRAPH ELEMENT
1,TITLE (HEAD)
2,SHAPED(BE FIT)
3,LEGEND(HAND AND FOOT)
4,UNIT (LIMIT)

27. example1 A B Car A is travelling 100km and it took it 10 hours
Car B is travelling 100km and it took it 20 hours A B
The greater the slope the greater the rate of change

28. Strategy of showing domain & range

29. Solution 1
We usually use to list the domain and range in a normal way. For instance:
The domain is (1,2,3,4)
The range is (1,2,3,4)

30. Solution 2
But since it is sometimes impossible to list all of the value, we can use these symbols to substitute them.<, >,< >
Domain; 0 x  4
Range 0 y 4

31. Solution 3
the use of interval notation is also a good idea, (the use of brackets）
Round brackets ( and ) mean the same thing as > and <
Square brackets [ and]mean the same as  and 
Domain; [0,4]
Range[0,4]

32. Graphing review Line: Continues forever in both directions
Line Segment: Has two endpoints
Ray: Has one endpoint and continues forever in the other direction
Open dot: Means this value is not included in the domain/range
Closed dot: Means this value is included in the domain/range
 is used to represent that all real number is available

33. You do
Domain (0,1] [1,3 ) (3,4]
Range y={0,1,2,3}

34. You do
Domain[ ]
Range [-3, ]

35. Continuous and Discrete Data

36. Continuous and Discrete Data
Continuous data refers to a linear graph that dots are connected into a line because it can be all value and measuring
Discrete data refers to a graph that dots are not connected because the elements are counting and integer
One people comes into the room for each hour
the car drives at a speed of 1 km each hour
Discrete data
Continuous data

37. Linear Relation

38. Linear and non-linear relation
(points in a straight line)
Non-linear
(points not a straight line)
Linear: means a graph has points that connect in a straight line or a graph is a straight line

39. Determine linear or not
Part 1
Justify in a table

40. Which represent a linear relation
Celsius(C)
Fahrenheit(F) 32 5 41 10 50 15 59 20 68 25 77
current(I)
watts(p) 5 75 10
300 15
675 20
1200 25
1875 +5 +9 +5
+75 +5 +9 +5
+225 +5 +9 +5
+375 +5 +9 +5
+525 +9 +5 +5
+675
Each increase 5
Each increase 9
Since the changes in both variables are constant , the table represent a linear relation.
The change in P is not constant so the table does not represent a linear relation

41. Determine linear or not
Part 2
Justify in order pairs

42. Which represent a linear relation
1,{0,75} {1,125} {2,175} {3,225}
There is a constant change in the first and second number so it is a linear system
2,{0,4} {1,9} {2,16} {3,25}
There is not a constant change in the first and second number so it is not a linear system

43. Determine linear or not
Part 3
Justify in a relation

44. y = -3x + 25 y = 2x2 + 5 linear non-linear
Conclusion: linear system are all unary functions, x do not have power