Functions3.3

is a function because each first element is paired with a different second element.

The relation

(2,3), (5,9), (1,0), (10, 2)A

(4,5), (4,0), ( 1,9)B

The relation

is NOT a function because (4, 5) and (4, 0) have the same first element and different second elements.

Which of the following are functions?

(9,0), (3,8), (5,8), (9, 1)

(0, ), (9, ), ( 2, ), ( 5, )

( 3,5), (7, 2), ( 3,2), (9, 1)

(0,1), (1,0), (1,1)

A

B t e q b

C

D

No

Yes

No

No

Vertical Line Test

There is a graphical method for deciding if a relation is a function. It’s called the vertical line

test. If a graph touches a vertical line more than once, at any place on the graph,

it is not a function.

Vertical Line Test – Does a vertical line touch two points of the graphed relation at once?

If so, it is not a function.

Example 6-1a

Determine whether the relation is a function. Explain.

Yes, it’s a function.

Determine whether the relation is a function. Explain.

x y–7 –12

–4 –9

2 –3

5 0

Yes, it’s a function.

Example 6-1c

Determine whether {(–5, 2), (–2, 5), (0, 7), (0, 9)} is a function. Explain.

No, it’s not a function.

Determine whether each relation is a function. Explain.

a.

Yes

b.

c. {(3, 0), (1, 2), (4, 0), (5, –1)}

–13

–42

–21

23

YX

No

Yes

Determine whether is a function.

No

Determine whether is a function.

Answer: yes

Determine whether this graph is a function.

Answer: yes

Example 6-2b

Determine whether this graph is a function.

Answer: no

The solid dot shows that (3, 2) belongs to the graph. The open dot shows that (3, 4) does not belong to the graph.

Write these answers in your notes: Does this graph represent a function?

Yes

Does this graph represent a function?

Yes

Does this graph represent a function?

Yes

Does this graph represent a function?

No

FunctionNotation

( , ) (1,4), (2,3), (3,2), (4, 8), (5,2)g x y

( , ) ( 4,0), (9,1), ( 3, 2), (6,6), (0, 2)h x y

Using Function NotationFind g(2) and g(5) for

.

g(2) = ____ g(5) = ____

Consider

What is h(9)? ___ h(6)?____ h(0)? _____

3 2

1 62

Now look away from your notes for a while.

Instead of y in 2 3y x we write f(x).

doesn’t mean f times x. ( )f x

It means the value of the function calledf at the point x.

( ) 2 3f x x

If find .

Multiply.

Subtract.

Answer:

Replace x with 4.

If find .

Multiply.

Subtract.

Answer:

Replace x with –5.

If find .

Distributive Property

Simplify.

Answer:

Replace x with 2 – x.

If find each value.

a.

b.

c.

Answer: 11

Answer: –11

Answer:

Write on your notes

If , find .

Multiply.

Simplify.

Answer:

Replace m with –3.

Answer: 8

If find each value.

Write on your notes

IfMultiple-Choice Test Item

A 69 . B 70. C 79. D 81.

Read the Test Item

The symbol is just a different notation for f(x). Solve the Test Item

Replace x with –5.

Answer: A

Think: .

Replace x with –5.

Simplify.

Answer: C

SAT Question

A 35. B 30. C 20. D 19.If

Finding the domain of a function from an equation:

What is the domain of f for ?4

( )3

xf x

x

Therefore, . That means that The domain of the functionis

3 0x 3.x

3x x

This implies that x is all real numbers except -3.

0

xAs we know, is undefined.

Try this. What is the domain of each?

( )( 1)( 3)

xg x

x x

( ) 3 9h x x

2 1( )

3 3

xp x

x

1, 3x x

x x

0x x

Classwork: 4, 18, 22/119-120

Get ready for a “Small Quiz” to be written

on your grade sheet.

The End

Quiz. Copy the problems and write the answer.

Put your grade paper on the front of your row, quiz side down.

1. Write the Cartesian Product A X B if A = {0, 1, 2} and B = {6, 7}.

2.Write the domain of A if A = {(7, 3), (-2,5), (7,1), (8, 9)}