Relations and Functions

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

Function VS Relation Part 1 Graph

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

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.

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

You do Hyperbola

Identify the following chart symbols function or not. Why?

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

Function VS Relation Part 2 Arrow Diagram

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

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

You do Which arrow diagrams represent functions?

Function VS Relation Part 3 Ordered pairs

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

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.

Independent and dependent variable

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

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

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

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

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

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

SKETCHING A GRAPH

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

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)

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

Strategy of showing domain & range

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)

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

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]

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

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

You do Domain[ ] Range [-3, ]

Continuous and Discrete Data

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

Linear Relation

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

Determine linear or not Part 1 Justify in a table

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

Determine linear or not Part 2 Justify in order pairs

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

Determine linear or not Part 3 Justify in a relation

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