Relations and Functions

9.2 Relations

9.5 Functions

F

E

Line Slope Y-intercept SI

Y=mx+b

SF

Ax + By=C

E

F

You may count the slope

or

Pick two points and use calculator

Ex 1

Find the distance between the points P(2, 2) and Q (4, -6).1, 1( )x y 2, 2( )x y

2 22 1 2 1( ) ( )d x x y y

2 2( ) ( )d 4 2 26

2 2( ) ( )d 2 8

d 4 64

68 d

Work inside parentheses

Do powers before doing the addition!!!!!!

Be careful Of Signs!!!

(neg)2 = POS.

68 d

) 2 10

) 68

) 2 10

) 2 17

A

B

C

D

8.2462116.32455532

8.246211

Ex 5

2 1midpoint2

x xx

2 1midpoint2

y yy

midpoint midpoint( , )x y

Find the midpoint whose endpoints are (1, -2) and (-17, 16)1, 1( )x y 2, 2( )x y

midpoint

2x

17 1

216

8

( , )8

midpoint

2y

16 2

214

7

7

Ex 8 M(-8, 7) is the midpoint of RS. If S has a coordinates (-6, 8),

find the coordinates of R.

2, 2( )x y

2

6 1x

16 x 16

( , )10

2

8 1y

6

,( )m mx y1, 1( )x y

R (x1, y1) S (-6, 8)M(-8, 7)

8 7(2) (2)

6 6

1x 10

(2) (2)

18 y 148 8

1y 6

Topics

Domain Range Mapping Inverse of a

Relation

Functions (relation/graphs)

Function Notation Evaluate Function

Relations (sets of ordered pairs)

Ways to display a relationList the set of ordered pairsGraphingTableMapping

X Y

2 2

5 7

-1 4

9 6

Table

Mapping

2 2

5 7

-1 4

9 6

4

2

-2

-4

-5 5

Graphing

2 2

5 7

-1 4

9 6

2 2

5 7

-1 4

9 6

2 2

5 7

-1 4

9 6

2 2

5 7

-1 4

9 6

Domain Range The domain of a

relation is the set of all the first coordinates from the ordered pairs.

The x values

The Range is the set of all the second coordinates from the ordered pairs.

The y values

A set of ordered pairs

{(2,-2), (-2,3), (5,7), (4,8)}

Domain

{ 2, -2, 5, 4}

Range

{-2, 3, 7, 8}

Ordered pairs

inparentheses

Sets denotedby brackets

Ex 1 List the ordered pairs, domain, range, and inverse

X Y

0 2

1 6

-5 -2

12 3

A set of ordered pairs{(0,2), (1,6), (-5,-2), (12,3)}

Domain{ 0, 1, -5, 12}

Range{2, 6, -2, 3}

Inverse{(2,0), (6,1), (-2, -5), (3, 12)}

What is domain?

X-values

What is Range?

y-values

Inverse---switch x and y

Ex 2 List the ordered pairs, domain, range, and inverse

X Y

8 -3

4 9

1 1

-13 7

A set of ordered pairs{(8,-3), (4,9), (1,1), (-13,7)}

Domain{ 8, 4, 1, -13}

Range{-3, 9, 1, 7}

Inverse{(-3,8), (9,4), (1,1), (7,-13)}

What is domain?

X-values

What is Range?

y-values

Inverse---switch x and y

Definition of a function

Function is a special relation where each “x” is paired with exactly one “y”

{(2,3) (4,5) (2,8)}

{(5,8), (2, 9), (3,9)}

Another way to say it X’s are girls y’s are boys Girls can’t cheat on a

boy but a boy can cheat on a girl

Not a function

Yes, a functionBoys can cheat

Examples 3-6

Which are functions?

#3 {(-9,4) (2,6) (3,8), (3,9)}

#4 {(4,6), (12, 9), (2,3), (4,6)}

#5 {(1,3) (14,15) (12,8), (14,5)}

#6 {(-5,8), (7, 8), (3,9)}

NO

YES

NO

YES

Vertical Line Test-- Can’t touch line more than once or not a function

Not a function

Yes! It’s a function

Not a function

Examples 7-10Examples 7-9

#7

______

#8

___

#8

_____YES

NO

YES

# 9

Function Notation

y=2x + 6 F(x) =2x + 6 Read “f of x” Represents the

value in the range for that particular value of x

F(3) is the function value for f for x=3.

Plug the 3 in for x F(3) = 2(3) + 6 F(3)= 6 + 6 = 12 Can write it as

(3,12)

Ex 10-12 2( ) 3g x x x

( ) 3 2f x x

10) ( 1)f

( 1) 3( 1) 2f

3 2

1

11) ( 3)g 112) ( )

3f

2( 3) 3( 3) ( 3)g 1 1( ) 3( ) 23 3

f

3(9) 3 1 2

327 3 24