Warm UpWarm Up

1. (5 3) 2 3

2. 3 (5 10) 2 4 (3 7)

3. 7 3

Evaluate k k let k

Solve x x x x

Graph x or x

Objective:1. Identify Domain and Range2. Know and use the Cartesian Plane3. Graph equations using a chart4. Determine if a Relation is a Function5. Use the Vertical Line Test for Functions

RelationsRelations A A relationrelation is a mapping, or is a mapping, or

pairing, of input values with output pairing, of input values with output values.values.

The set of input values is called the The set of input values is called the domaindomain..

The set of output values is called The set of output values is called the the rangerange..

Domain & RangeDomain & Range

Domain is the set of all x values.

Range is the set of ally values.

Example 1:

Domain- D: {1, 2} Range- R: {1, 2, 3}

{(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)}

Example 2: Example 2:

Find the Domain and Range of the following relation:

{(a,1), (b,2), (c,3), (e,2)}

Domain: {a, b, c, e} Range: {1, 2, 3}

Page 107

3.2 Graphs3.2 Graphs

Cartesian Coordinate SystemCartesian Coordinate System Cartesian coordinate planeCartesian coordinate plane x-axisx-axis y-axisy-axis originorigin quadrantsquadrants

Page 110

A Relation can be represented by a A Relation can be represented by a set of set of orderedordered pairspairs of the form (x,y) of the form (x,y)

Quadrant IX>0, y>0

Quadrant IIX<0, y>0

Quadrant IIIX<0, y<0

Quadrant IVX>0, y<0

Origin (0,0)

Plot:(-3,5) (-4,-2) (4,3) (3,-4)

Every equation has solution points (points which satisfy the equation).

3x + y = 5

(0, 5), (1, 2), (2, -1), (3, -4) Some solution points:

Most equations have infinitely

many solution points.

Page 111

Ex 3. Determine whether the given ordered pairs are solutions of this equation.

(-1, -4) and (7, 5); y = 3x -1

The collection of all solution points is the graph of the equation.

Ex4 . Graph y = 3x – 1.

x 3x-1 y

Page 112

Ex 5. Graph y = x² - 5

x x² - 5 y

-3

-2

-1012

3

What are your What are your questions?questions?

FunctionsFunctions

Page 116Page 116

•A relation as a A relation as a functionfunction provided there is provided there is exactly one output for each input.exactly one output for each input.

•It is It is NOTNOT a function if at least one input has a function if at least one input has more than one outputmore than one output

INPUT

(DOMAIN)

OUTPUT (RANGE)

FUNCTIONMACHINE

In order for a relationship to be a function…

EVERY INPUT MUST HAVE AN OUTPUT

TWO DIFFERENT INPUTS CAN HAVE THE SAME OUTPUT

FunctionsFunctions

ONE INPUT CAN HAVE ONLY ONE OUTPUT

Example 6

No two ordered pairs can have the No two ordered pairs can have the same first coordinatesame first coordinate

(and different second coordinates).(and different second coordinates).

Which of the following relations are functions?

R= {(9,10, (-5, -2), (2, -1), (3, -9)}

S= {(6, a), (8, f), (6, b), (-2, p)}

T= {(z, 7), (y, -5), (r, 7) (z, 0), (k, 0)}

Identify the Domain and Range. Then Identify the Domain and Range. Then tell if the relation is a function.tell if the relation is a function.

Input Output

-3 3

1 1

3 -2

4

Domain = {-3, 1,3,4}Range = {3,1,-2}

Function?Yes: each input is mappedonto exactly one output

Input Output

-3 3

1 -2

4 1

4

Identify the Domain and Range. Then tell if the relation is a function.

Domain = {-3, 1,4}Range = {3,-2,1,4}

Function?No: input 1 is mapped onto Both -2 & 1

Notice the set notation!!!

Look at example 1 on page 116Look at example 1 on page 116

Do “Try This” a at the bottom of page 116Do “Try This” a at the bottom of page 116

1. {(2,5) , (3,8) , (4,6) , (7, 20)}

2. {(1,4) , (1,5) , (2,3) , (9, 28)}

3. {(1,0) , (4,0) , (9,0) , (21, 0)}

The Vertical Line TestThe Vertical Line TestIf it is possible for a vertical line

to intersect a graph at more than one point, then the graph is NOT the graph of a function.

Page 117

(-3,3)(4,4)

(1,1)

(1,-2)

Use the vertical line test to visually check if the relation is a function.

Function?No, Two points are on The same vertical line.

(-3,3)

(4,-2)

(1,1) (3,1)

Use the vertical line test to visually check if the relation is a function.

Function?Yes, no two points are on the same vertical line

ExamplesExamples

I’m going to show you a series of I’m going to show you a series of graphs.graphs.

Determine whether or not these Determine whether or not these graphs are functions.graphs are functions.

You do not need to draw the graphs in You do not need to draw the graphs in your notes.your notes.

#1 Function?

Function?#2

Function?#3

Function?#4

Function?Function?#5

#6 Function?

Function?#7

Function?#8

#9 Function?

Function?Function?#10

Function?#11

Function?#12

)(xf“f of x”

Input = x

Output = f(x) = y

Function Function NotationNotation

y = 6 – 3x

-2

-1

0

1

2

12

9

6

0

3

x y

f(x) = 6 – 3x

-2

-1

0

1

2

12

9

6

0

3

x f(x)

Before… Now…

(x, y)

(input, output)

(x, f(x))

Find Find gg(2) and (2) and gg(5).(5).

g = {(1, 4),(2,3),(3,2),(4,-8),(5,2)}

g(2) = 3 g(5) = 2

Example 7

Consider the functionConsider the function h= { (-4, 0), (9,1), (-3, -2), (6,6), (0, -2)}h= { (-4, 0), (9,1), (-3, -2), (6,6), (0, -2)}

Example 8

Find h(9), h(6), and h(0).

Example 9. Example 9.

f(x) = 2x2 – 3

Find f(0), f(-3), f(5a).

F(x) = 3xF(x) = 3x22 +1 +1Example 10.

Find f(0), f(-1), f(2a).

f(0) = 1

f(-1) = 4

f(2a) = 12a2 + 1

The set of all real numbers that you can plug into the function.

DomainDomain

D: {-3, -1, 0, 2, 4}

f :{( , ), ( , ), ( , ), ( , ), ( , )} 3 0 1 4 0 2 2 2 4 1

g(x) = -3x2 + 4x + 5

D: all real numbers

Ex.

Ex.

What is the domain?What is the domain?

f xx

x( )

4

3x + 3 0

x -3

D: All real numbers except -3

h xx

( )1

5x - 5 0

Ex.

What is the domain?What is the domain?

D: All real numbers except 5

D: All Real Numbers except -2

Ex.

x + 2 0f xx

( )

1

2

What are your What are your questions?questions?