Functions Basic Properties. 7/9/2013 Functions 2 Dow Jones Closing Average Daily closing stock prices for a week Functions in Everyday Life Question:

Functions

Basic Properties

7/9/2013 Functions2

Dow Jones Closing Average Daily closing stock prices for a week

Functions in Everyday Life

Question:

What is the formula for the function ? Does this graph represent a function ?

8149

8700

8516

8332

M T W Th F

Stock Price in Dollars

7/9/2013 Functions33

Average daily high temperature Historical daily data points for July


Question:

What is the formula for the function ? Does this graph represent a function ?

1 10 20 3190

95

100

105

110Daily High (F°)


PC Color Palette Table lookup


Question:

What is the graph ?

Does this table represent a function ?

red orn yel chr grn ..... blu dkb mag ….

1 22 31 38 43 ..... 207 233 245 ….Index

Color


Relational Database Table Student data functionally related to id


Id No Last Name First Name

1396 Miller James1372 Burrows Susan

1448 Wilson Dorothy1531 Bronson Charles .... .... ....

KEY ATTRIBUTESQuestion:

What is the graph ?

Does this table represent a function ?



Function with formula and graph Travel time as a function of speed

t =dr

r

t

Question:

Is the function the graph… or the formula … or something else ?

7/9/2013 Functions7


Function with formula and graph Conversion of Fahrenheit to Centigrade

( )C =59

F– 32 F

C

212

100

32 ●

●

Question:

Is the function the graph… or the formula … or something else ?


Functions in Everyday Life Let’s Review

Functions with graphs but no formula DOW Jones Closing Average Average daily temperatures

Functions with no graph, no formula PC Color Palette Student relational database table



Let’s Review

Functions with a formula and a graph

Travel time based on rate of travel Conversion of Fahrenheit to Centigrade

Question:How do we define all of these … in one simple way ?


Functions in Everyday Life Function Characterization

What is the common characteristic ?

Ordered Pairs:(day, average),

(day, temperature F), (student id, name),

(speed, travel time),

(F, C)

Relates one set of data with another


Ordered Pairs

Ordered Pair Composed of two components:

First Component and Second Component

( a , b )

First Component

Second Component

Component Types:Can be any kind of objects


Ordered Pairs

Sets of Ordered Pairs

Notation:

Example: Colors

Ordered Pair:

{ ( a , b ) , ( c , d ) , ( e , f ) , ( g , h ) }

Set of ordered pairs:

{ ( orange , blue ) , ( red , green ) , … }

( red , green )


Ordered Pairs

Example:

Numbers

Ordered pair:


{ ( -6 , 10 ) , ( 3 , 0 ) , ( 2 , 7 ) , ( -2 , 0 ) }

( 3 , 0 )

Question: Does the order of the pairs matter ?


Ordered Pairs

Numbers and Colors

Ordered pair:


{ ( 0 , black ) , ( 1 , red ) , ( 2 , yellow ) … }

( 3 , blue )

Example:

Question: Does this appear to be a function ?


Ordered Pairs

Historic Figures

Ordered pair:


{ ( 0123 , Jim Bowie ), ( 0124 , Anson Jones) … }

( 0129 , Sam Houston )

Example:

Question: How many entries can this set have ?


{ … , , … }

Functions Functional Relationships

Ordered pairs are the key

Set A Set B

a b( , )a b

( , )a bS =

Question: Is S a relation ? YESIs S a function ? Maybe

… it depends


{ }

Functions Function

Relates each member of the domain with exactly one member of the range

domain rangea b

c d

( , )a bS =

Question: Is S a function ? YES … probably

j k

, …( , )j k,( , )c d,


domain

range

Functions Function

Maps each domain element to exactly one range element

a bc d

Question: Is S a function ?

j k

Is (c , b) OK ?YES !

Are (c , b) and

(c , k) OK ?NO !

Not with (c , b) and (c , k) !!

{ }( , )a bS = , …( , )c k,( , )c b,


Functions Function

We know what a function does …

… but what is a function ?

Definition:

A function is a set of ordered pairs, no two of which have the same first component.


Functions

Function Alternate Definition (Textbook):

A function is a relation in which each domain element is related to exactly one range element.


What’s A Function?

Class Definition of Function

A function is a set of ordered pairs,

no two of which have the same first component.

Note :

The definition is a complete sentence The definition names the thing it

defines: function


Functions

Examples 1. A = { (b, d), (g, h), (x, y) }

2. B = { (y, a), (y, b), (y, c), (y, d) }

3. C = { (a, y), (b, y), (c, y), (d, y) }

4. D = { (1, 3), (3, 1), (1, 1), (3, 3) }

Which of these are functions ?

Which are not ? 1 and 3

2 and 4 WHY ?


Describing Functions

Functions as finite sets – list notation Examples

A = { ( a , b ) , ( c , d ) , ( e , f ) , ( g , h ) }

B = { ( -6 , 10 ) , ( 3 , 0 ) , ( 2 , 7 ) , ( -2 , 0 ) }

C = { }( org , blu ) , ( red , grn ) , ( blk , brn )

Question:Any two pairs with same first component?



Functions as large sets Large sets too big to list

Example:

– set builder

F is the set of all ordered pairs (x,y)such that y = 2x for all real x

F = { }(x, y) | y = 2x , x is any real number

– maybe infinite



Set builder examples

G = { (x, y) y = 2x , x > 0 }

S = { (x, y) x = student-ID, y a student }

D = { (d, c) d = day, c = DOW Jones close }

Do all functions have formulas?

Do functions always have graphs?

Question:


Describing Functions Do functions always have graphs?

No, but ordered pairs make graphing easy

F = { ( -5 , 4 ) , ( -2 , 1 ) , ( 2 , 7 ) , ( 4 , 1 ) }

a

b

(-5, 4)

(-2, 1)

(2, 7)

(4, 1)

What is the graph ?

… and now?Scatterplot

Line graph

Question:


y

Function Graphsf = { (x , y) | y = 2x – 1 }

x

(-1, -3)

(1, 1)

(2, 3)

(3, 5)

(4, 7)

The Graph of f

•

•

•

•

•


Domain = [ 0, 4 ]

Finding Domain and RangeConsider: f = { (x , y) | y = 2x – 1 , 0 x 4 }

x

y

(0, -1)

(4, 7) The Graph of f

Range = [ -1, 7 ]

Note: For THIS function, the domain and range are closed intervals

= { x | 0 x 4 }

= { y | -1 y 7}

•

•


The Language of Functions

Notation f = { (x, y) y = 2x , x is any real number }

f is the name of the function

f(x) is the value of the function at x

{ (x, y) y = 2x , x is any real number }

is the function itself


The Language of Functions

Notation f = { (x, y) y = 2x , x is any real number }

x is the independent variable (or input)

y is the dependent variable (or output)

Input determines output

We say y is a function of x

… OR … output is a function of input


Functions as Tables

Example Tuition at John Q. Public

Junior College is $250 per semester hour for total hours less than 12

For 12 or more hours the tuition is a flat charge of $2800

For hours H and tuition T construct a table for a variety of course loads up to 18 hours

T H

750 3

1500 6

2250 9

2800 12

2800 15

2800 18


Functions as Tables

Example T H

750 3

1500 6

2250 9

2800 12

2800 15

2800 18

Question:

Question:

Is H a function of T ?

Is T a function of H ?

Does T determine H ?

Does H determine T ?

H is NOT a function of T

T is IS a function of H

Question:

Does this last fit the definition ?

No

Yes

So …

So …


Graphs, Sets and Tables

Ordered Pairs and Functions

Input

Output

x

y●

●

●

(x, y)

Ordered Pair

Input Output

x y

f = { y = 2x + 1 }(x, y)

(x, y)

Table


Graph and Functions

Vertical Line Test No vertical line touches the graph of a function at more than one point

Function ?YES

Function?NO

Function?YES

●● ●

●

●

● ●

●(x1, y1)

(x1, y2)

x = x1


Graph and Functions

Vertical Line Test ExamplesDoes any vertical line cut the graph more than once ?

Function?YES

Function?NO

Function?YES

x

y

x

y

x

y

●●

●●

●●

y = x2 x = y2

x = y2, y > 0


Set of all second components

Set of all first components

Relation As A Set

{ }( a , b ) , ( c , d ) , ( e , f ) , ( g , h )

Domain:

Range:

{ a , c , e , g }

{ b , d , f , h }

NOTE:

DOMAIN and RANGE are SETS of objects

Relation: Any set of ordered pairs

Notaton:


Examples

Relations A = { ( a , b ) , ( c , d ) , ( c , f ) , ( g , b ) }

Domain of A =

Range of A =

{ a, c, g }

{ b, d, f }


Relations B = { ( -6 , 7 ) , ( 3 , 0 ) , ( 2 , 7 ) , ( -2 , 0 ) }

Domain of B =

Range of B =

{ -6, -2, 2, 3 }

Examples

{ 0, 7 }


Examples

Relations S = { (1493, Bob), (3872, Sally), (2840, Bob),

(5492, Mary) }

Domain of S =

Range of S =

{ 1493, 3872, 2840, 5492 }

{ Bob, Sally, Mary }


The Word “RANGE”

The two meanings of RANGE

Measure of dispersion of 1-variable data Maximum-minimum difference A real number , a statistic

Target for a relation Derived from ordered pairs Set of all second components


Graphical Presentations

One-variable data in one dimension

Example:

-21, -13, -8, 3, 5, 9, 12, 22

0 10 20-20 -10

A plot of numbers


Graphical Presentations Two-variable data in two dimensions

Example:

x y

-7 2

-5 -3

-1 4

2 -4

5 -2

8 6

(-7,2)

(-5,-3)

(-1,4)

(2,-4)(5,-2)

(8,6)

x

yA plot of ordered pairs


Think about it !

Documents

Functions Basic Properties. 7/9/2013 Functions 2 Dow Jones Closing Average Daily closing stock prices for a week Functions in Everyday Life Question: