Functions 3.5 An Introduction to Funtions. Independent and Dependent Variables

Functions 3.5

An Introduction to Funtions

Independent and Dependent Variables

Independent and Dependent VariablesDefinitions

Indepdent Variable: the variable that someone/something changes directly

Dependent Variable: the result of the independent variable changing


independent(cause) dependent(effect)

Examples:

ind: hours I study for class

dep: grade in class


ind: hours I worked in a pay period

dep: how much I got paid


ind: amount of time spent on the treadmill this week

dep: blood pressure


There is a relationship between the independent and dependent variable

This relationship can be shown using an ordered pair

(x,y)

(cause, effect)

Functions

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)

Functions

You can create your own function

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

● This is a function because . . . ?

Functions

Function or Not a Function

Is H a function? Why or Why not?

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

Functions


Is H a function? Why or Why not?

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

Functions


Is G a funtion?

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

Functions


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 FunctionsAssume 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 customers depends on people walking through the front door.

Only greet customers when someone enters!!!

Real World Functions

Usher at the movie

When ever a movie end, clean the theater

U = (Movie Ends, Clean Theater)

cleaning of the theater depends on the moving ending.

Only clean the theater when the movie ends!

Real World Functions

Any other examples of functions?

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.

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.

Functions

Function of Not a Function?

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


Why?

Functions

Function of Not a Function?

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


Why?

Functions

Function of Not a Function

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

Why?

Functions


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

Why?

Function

Functions


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

Why?

Functions


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

Why?

Not a Function

Functions


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

Why?

Functions


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

Why?

Not a Function

Functions


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

Why?

Functions


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

Why?

Function

Domain and Range

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 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 Range

What is the Domain and Range?

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

Domain?

Range?

Domain and Range


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

Domain {1, 4, 10}

Range {-2, 7, 16, }

Domain and Range


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

Domain?

Range?

Domain and Range


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

Domain {-7, -5, 6}

Range {-18 ,-1 31}

Domain and Range


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

Domain?

Range?

Domain and Range


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

Domain {-4, 0, 4}

Range {-3,0}

Domain and Range


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

Domain?

Range?

Domain and Range


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

Domain {1, 2, 3}

Range {3}

Finding Domains and Ranges From a Graph


● What is the domain and range?

● Domain {-1, 1, 2 }● Range {-1, 1, 2}


● Try on your own.● What is the domain

and range● Domain● Range


● Try on your own.● What is the domain

and range● Domain {-4, -2, 3}● Range {-2, -1, 1}


●What is the domain and Range of this graph?●Domain?●Range?


●Try on your own●What is the domain and Range of this graph?●Domain [-2, 2]●Range [-1, 4]


Try on your own●What is the domain and Range of this graph?

●Domain?

●Range?

●Finding Domains and Ranges From a Graph

● What is the domain and range for this graph

● Arrow heads mean continue indefinitely

● Domain(-∞, ∞)

● Range (-∞, ∞)


Try This One

What is the domain and range for this graph?

Arrow heads mean continue indefinitely

Domain(-∞, ∞)

Range (-∞, ∞)




Domain(-∞, ∞)

Range [-1, ∞)


Try this One



Domain(-∞, ∞)

Range (- ∞, 4]

Vertical Line Test

●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


● Any vertical line drawn here will only cross once.

● There fore it passes the vertical line test

●Vertical line test

● Does This Pass the vertical line test

● Function or Not


● This does not pass the vertical line test

● Most lines would cross this graph more than once

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Functions 3.5 An Introduction to Funtions. Independent and Dependent Variables