2-1: Relations and FunctionsAlgebra 2

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

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

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

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

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

The Vertical Line TestUsed with graphs

◦If the vertical line touches the graph in more than one point, the graph is not a function.

Why the Vertical Line Test Works

1,x y

2,x y

(3, )

(3, )

4

-4

Homework: Day 1 p. 65 #11, 13-16, 20, 22, 23, 26, 28, 45-51 odd

Euhler’s Function Notation:

Mapping Notation:

Types of Function Notation

: T x x

)(xfy

: 3 f n n

)(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

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

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

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

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.

Homework: Day 2 p. 65 #8-10, 12, 17-19, 25, 29, 30, 32, 39-43, 44-52 even