2-1: relations and functions algebra 2. what is a relation a set of inputs and outputs can be...
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2-1: Relations and FunctionsAlgebra 2
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What is a RelationA set of inputs and outputsCan be represented in 4 different
ways:Ordered
PairsMapping Diagram
Table of Values
Graph
1,94, 21, 119,10
1
4
9
92
1110
Inputx
Outputy
1 9
1 -11
4 -2
9 10
10
5
-5
-10
10
B
C
A
D
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In 2000, the 4 most populous states(in millions), were CA {32}, TX {21}, NY {19} and FL {16}. The numbers of U.S. Representatives were CA {53}, TX {32}, NY {29} and FL {25}. How can you represent a relation for these data in 4 different ways?
Ordered Pairs
Mapping Diagram
Table of Values
Graph
32,5321,3219,2916,25
32211916
53322925
Inputx
Outputy
32 53
21 32
19 29
16 25
80
60
40
20
20 40
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Domain and Range of a Relation
Domain RangeSet of inputsx coordinates of
ordered pair
Set of outputsy coordinates of
ordered pair
x 1 2 3 4 5 6
y $5.00$10.0
0$15.0
0$20.0
0$25.0
0$30.00
Domain: _______________________________
Range: _________________________________
1,2,3,4,5,6
5.00,10.00,15.00,20.00,25.00,30.00
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Identifying FunctionsWhat is a Function?
◦A relation in which each element of the domain corresponds with one and only one element in the range.
32211916
53322925
1,9 , 4, 2 , 1, 11 , 9,10
Look for duplicates in the DomainFUNCTION
NOT AFUNCTION
Complete Got It? p. 62 #3
1).
2).
No Yes
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Components of Functions
Independent Variable Dependent Variable
Solution depends on it.
Input
Usually the ______ variable in a table
first
Depends on the first variable for a value.
Output
Usually the _______ variable in a table
second
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The Vertical Line TestUsed with graphs
◦If the vertical line touches the graph in more than one point, the graph is not a function.
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Why the Vertical Line Test Works
1,x y
2,x y
(3, )
(3, )
4
-4
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Homework: Day 1 p. 65 #11, 13-16, 20, 22, 23, 26, 28, 45-51 odd
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Euhler’s Function Notation:
Mapping Notation:
Types of Function Notation
: T x x
)(xfy
: 3 f n n
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)(xfy • Notations are used as a short hand
• Used mainly with sentences (equations)
• Letter denotes the name of the function
103 xxf )(•Value of the function
•Dependent variable
•Output
•Argument of the function
•Independent variable
•Input
Euhler’s Notation:
3 10y x Equation
Function Rule
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Ex.1: Here, the three distances (thinking, braking, stopping) are each a function of speed. Rewrite as Function Rules using Euhler’s f(x).
Thinking distance at speed x: y= x.
Braking distance at speed x:
Stopping distance at speed x:
2
20
xy
2
20
xy x
( )T x x
2
( )20
xB x
2
( )20
xS x x
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Using Function Notation
xInput
Function RuleOutput
-4
0
0.2
For , what is the output for the input -4, 0 and 0.2?
3 2f x x
3 2f x x f x
4 3 4 2f
0 3 0 2f
0.2 3 0.2 2f
14
2
1.4
Complete Got It? p. 63 #5
a). 9 b). 1 c). -19
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A pizza costs $14; the flat delivery fee is $1.50. What function rule models the total cost of the number of pizzas delivered? Evaluate for 5 pizzas.
Relate
Define
Write
Total Cost
Cost per
Pizza
# of Pizzas
Delivery Fee
# of pizzasp Total costC p
14 1.5C p p
C p 14 p 1.5
5C 14 5 1.5 71.5
The total cost for 5 pizzas is $71.50
Evaluate
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Complete Got It? p. 64 #6
1.19C x x
15C 1.19 15 17.85
Let x = # of bottles purchased
Let C = Total cost
Complete Lesson Check p. 64 # 1-4
1).
2).
3).
4).
0,3,4D 2,1,2,4R
4, 3,0,4D 4, 3,0,4R
No,
3 has two values in the range.
Yes,
each value in the domain has only one value in the range.
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Homework: Day 2 p. 65 #8-10, 12, 17-19, 25, 29, 30, 32, 39-43, 44-52 even