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Functions 3.5. An Introduction to Funtions. Independent and Dependent Variables. Independent and Dependent Variables. Definitions Indepdent Variable: the variable that someone/something changes directly Dependent Variable: the result of the independent variable changing.

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functions 3 5
Functions 3.5

An Introduction to Funtions

independent and dependent variables3
Independent and Dependent Variables

Definitions

Indepdent Variable: the variable that someone/something changes directly

Dependent Variable: the result of the independent variable changing

independent and dependent variables4
Independent and Dependent Variables

independent(cause) dependent(effect)

Examples:

ind: hours I study for class

dep: grade in class

independent and dependent variables5
Independent and Dependent Variables

ind: hours I worked in a pay period

dep: how much I got paid

independent and dependent variables6
Independent and Dependent Variables

ind: amount of time spent on the treadmill this week

dep: blood pressure

independent and dependent variables7
Independent and Dependent Variables

There is a relationship between the independent and dependent variable

This relationship can be shown using an ordered pair

(x,y)

(cause, effect)

functions9
Functions

Function: a relation in which, for each value of the first component there is exactly one value of the second component

Example: Remote control

(x,y)

(cause, effect)

functions10
Functions

You can create your own function

F={(1,1), (2,2),(3,3),(4,4)}

This is a function because . . . ?

functions11
Functions

Function or Not a Function

Is H a function? Why or Why not?

H={(4,12),(3,7),(5,23)}

functions12
Functions

Function or Not a Function

Is H a function? Why or Why not?

H={(4,12),(3,7),(5,23)}

functions13
Functions

Function or Not a Function

Is G a funtion?

G={(1,1),(2,5),(3,7),(4,1)}

functions14
Functions

Function or Not a Function

Is G a funtion?

G={(1,1),(2,5),(3,7),(4,1)}

Look closely at G

You can have the same out put for different inputs

Just as on your computer you can have two buttons(input) do the same action(output)

Example: (Shift Key)

real world functions
Real World Functions

Assume You own a resturant

You require the host to always say "Welcome. How many will be in your party today?", every time someone walks through the door.

This is a function (input, output)

H = ([Person enters door], [Greeting])

Greeting customersdepends on people walking through the front door.

Only greet customers when someone enters!!!

real world functions16
Real World Functions

Usher at the movie

When ever a movie end, clean the theater

U = (Movie Ends, Clean Theater)

cleaning of the theaterdepends on the moving ending.

Only clean the theater when the movie ends!

real world functions17
Real World Functions

Any other examples of functions?

functions18
Functions

Function only if for every input there is one output

D = {(2,5), (3,4), (1,8) (2,7), (4,5)}

Why is this not a function?

Think back to remote control.

functions19
Functions

Function only if for every input there is one output

D = {(2,5), (3,4), (1,8) (2,7), (4,5)}

Why is this not a function?

Think back to remote control.

functions20
Functions

Function of Not a Function?

{(1,2),(3,4),(5,6)}

Function or Not a Function

Why?

functions21
Functions

Function of Not a Function?

{(1,2),(3,4),(5,6)}

Function or Not a Function

Why?

functions22
Functions

Function of Not a Function

{(1,2), (2,2), (3,2), (4,2), (5,2) }

Why?

functions23
Functions

Function of Not a Function

{(1,2), (2,2), (3,2), (4,2), (5,2) }

Why?

Function

functions24
Functions

Function of Not a Function

{(1,2), (2,2), (3,2), (4,7), (3,8) }

Why?

functions25
Functions

Function of Not a Function

{(1,2), (2,2), (3,2), (4,7), (3,8) }

Why?

Not a Function

functions26
Functions

Function of Not a Function

{(2,4), (6,8), (10,12), (14,16), (18,20) }

Why?

functions27
Functions

Function of Not a Function

{(2,4), (6,8), (10,12), (14,16), (6,20) }

Why?

Not a Function

functions28
Functions

Function of Not a Function

{(1,4), (2,8), (12,12), (7,4), (8,8) }

Why?

functions29
Functions

Function of Not a Function

{(1,4), (2,8), (12,12), (7,4), (8,8) }

Why?

Function

domain and range31
Domain and Range

The domain is set of all values of the independent variable(x)

The range is the set of all values of the dependent variable(y)

domain and range32
Domain and Range

Example

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

Domain is{1,3,5,7} First Component of the ordered pair

Range is {2,4,6,8} Second Component of the ordered pair

domain and range33
Domain and Range

What is the Domain and Range?

{(1,7), (10,16), (4,-2)}

Domain?

Range?

domain and range34
Domain and Range

What is the Domain and Range?

{(1,7), (10,16), (4,-2)}

Domain {1, 4, 10}

Range {-2, 7, 16, }

domain and range35
Domain and Range

What is the Domain and Range?

{(-5, 31), (6,-1), (-7,-18)}

Domain?

Range?

domain and range36
Domain and Range

What is the Domain and Range?

{(-5, 31), (6,-1), (-7,-18)}

Domain {-7, -5, 6}

Range {-18 ,-1 31}

domain and range37
Domain and Range

What is the Domain and Range?

{(0 -3), (4,-3), (-4,0)}

Domain?

Range?

domain and range38
Domain and Range

What is the Domain and Range?

{(0 -3), (4,-3), (-4,0)}

Domain {-4, 0, 4}

Range {-3,0}

domain and range39
Domain and Range

What is the Domain and Range?

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

Domain?

Range?

domain and range40
Domain and Range

What is the Domain and Range?

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

Domain {1, 2, 3}

Range {3}

finding domains and ranges from a graph42
Finding Domains and Ranges From a Graph

What is the domain and range?

Domain {-1, 1, 2 }

Range {-1, 1, 2}

finding domains and ranges from a graph43
Finding Domains and Ranges From a Graph

Try on your own.

What is the domain and range

Domain

Range

slide44

Finding Domains and Ranges From a Graph

  • Try on your own.
  • What is the domain and range
  • Domain {-4, -2, 3}
  • Range {-2, -1, 1}
finding domains and ranges from a graph45
Finding Domains and Ranges From a Graph

What is the domain and Range of this graph?

Domain?

Range?

slide46

Finding Domains and Ranges From a Graph

  • Try on your own
  • What is the domain and Range of this graph?
  • Domain [-2, 2]
  • Range [-1, 4]
finding domains and ranges from a graph47
Finding Domains and Ranges From a Graph

Try on your own

What is the domain and Range of this graph?

Domain?

Range?

finding domains and ranges from a graph48
Finding Domains and Ranges From a Graph

What is the domain and range for this graph

Arrow heads mean continue indefinitely

Domain(-∞, ∞)

Range (-∞, ∞)

slide49

Finding Domains and Ranges From a Graph

Try This One

What is the domain and range for this graph?

Arrow heads mean continue indefinitely

Domain(-∞, ∞)

Range (-∞, ∞)

finding domains and ranges from a graph50
Finding Domains and Ranges From a Graph

What is the domain and range for this graph?

Arrow heads mean continue indefinitely

Domain(-∞, ∞)

Range [-1, ∞)

slide51

Finding Domains and Ranges From a Graph

Try this One

What is the domain and range for this graph?

Arrow heads mean continue indefinitely

Domain(-∞, ∞)

Range (- ∞, 4]

vertical line test53
Vertical Line Test

The vertical line test determines if a graph is a function

Use: draw a vertical line anywhere on the graph.

If it crosses the graph more than once it is not a function

slide54

Vertical Line Test

  • Any vertical line drawn here will only cross once.
  • There fore it passes the vertical line test
vertical line test55
Vertical line test

Does This Pass the vertical line test

Function or Not

vertical line test56
Vertical Line Test
  • This does not pass the vertical line test
  • Most lines would cross this graph more than once