FUNCTIONS & GRAPHSFUNCTIONS & GRAPHS

FUNCTIONS & NOTATIONFUNCTIONS & NOTATIONWEEK 10WEEK 10

Functions and Graphs (15 hours)Functions and Graphs (15 hours)

The Subtopics:1. The concept of a function and its

notation (2 hrs)2. Graphs of functions (6 hrs)3. Composite functions (2 hrs)4. Inverse functions (2 hrs)5. Limits and continuity (3 hrs)

Functions and NotationsFunctions and Notations

Important Ideas:1. Concept of a function2. Meanings of domain3. Meaning of codomain4. Meaning of range5. The equality of two functions6. One-to-one functions7. Many-to-one functions

Some Net ResourcesSome Net Resources

Slides have been adapted for use from some publiclySlides have been adapted for use from some publiclyavailable resources. For further information, check outavailable resources. For further information, check outthe following sitesthe following sites http://teachers.henrico.k12.va.us/math/hcpsalgebra2/Documents/2http://teachers.henrico.k12.va.us/math/hcpsalgebra2/Documents/2--1/21/2--

1RelationsFunctions.ppt1RelationsFunctions.ppt http://www.csie.ndhu.edu.tw/~rschang/dmchap5.ppthttp://www.csie.ndhu.edu.tw/~rschang/dmchap5.ppt http://www.mathxtc.com/Downloads/Tutorials/Year%2010/Relations%2http://www.mathxtc.com/Downloads/Tutorials/Year%2010/Relations%20and%200and%20

Functions.pptFunctions.ppt http://www.mgt.ncu.edu.tw/~ylchen/dismath/chap05.ppthttp://www.mgt.ncu.edu.tw/~ylchen/dismath/chap05.ppt http://www.animask12.net/vrichardson/prealgebra/ppt/Prehttp://www.animask12.net/vrichardson/prealgebra/ppt/Pre--

Algebra%20Lesson%208Algebra%20Lesson%208--1.ppt1.ppt http://hillgrovehighschool.typepad.com/files/functionshttp://hillgrovehighschool.typepad.com/files/functions--relationsrelations--1.ppt1.ppt

RelationsRelationsAA relationrelation is a mapping, or pairing, ofis a mapping, or pairing, of

input values with output values.input values with output values.

The set of input values is called theThe set of input values is called thedomaindomain..

The set of output values is called theThe set of output values is called therangerange..

Domain & RangeDomain & Range

Domain is the set ofall 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)}

Do not repeat the values for the domain andrange

Example 2:Example 2:

Find the Domain and Range of thefollowing relation:

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

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

Concept of a Function

•• A relation is aA relation is a functionfunction provided there isprovided there isexactly one output for each input.exactly one output for each input.

•• It isIt is NOTNOT a function if at least one input hasa function if at least one input hasmore 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 THESAME OUTPUT

ONE INPUT CAN HAVE ONLYONE OUTPUT

Example 6: Which of the following relations arefunctions?

No two ordered pairs can have theNo two ordered pairs can have thesame first coordinatesame first coordinate

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

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. ThenIdentify the Domain and Range. Thendetermine if the relation is a function.determine 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. Thentell if the relation is a function.

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

Function?No: Input 1 is mapped ontoboth -2 & 1

Input Output

-3 3

1 -2

4 1

5 4

Identify the Domain and Range. Thentell if the relation is a function.

Domain = {-3, 1, 4, 5}Range = {3, 1, 4}

Function?No: Input 5 is notmapped onto any output

Domain, Range & Codomain of aFunction

Domain = {-3, 1, 4, 5}

Range = {3, 1, 4}

Codomain = {3, -2, 1, 4}

Input Output

-3 3

1 -2

4 1

5 4

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 morethan one point, then the graphis NOT the graph of a function.

(-3,3)(4,4)

(1,1)

(1,-2)

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

Function?No, Two points are onthe same vertical line.

(-3,3)

(4,-2)

(1,1) (3,1)

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

Function?Yes, no two points areon the same vertical line

AppyingAppying the vertical line testthe vertical line test

Determine whether or not theseDetermine whether or not thesegraphs are functions.graphs are functions.

You do not need to draw the graphs inYou do not need to draw the graphs inyour notes.your notes.

#1 Function?

Function?#2

Function?#3

Function?#4

Function?Function?#5

#6 Function?

Function?#7

Function?#8

#9 Function?

“f of x”

Input = x

Output = f(x) = y, also called theimage of value of the function

f: x yf (x)

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)

One way of representinga linear function…

The alternative way …

(x, y)

(input, output)

(x, f(x))

FindFind 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

The value of a function

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

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

F(x) = 3xF(x) = 3x22 +1+1

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

f(0) = 1

f(-1) = 4

f(2a) = 12a2 + 1

f(x) = 2x2 – 3

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

Further ExamplesFurther Examples

The set of all real numbers that youcan 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

What is the domain?What is the domain?

f xxx

( )

43

x + 3 0x -3

D: All real numbers except -3

h xx

( ) 1

5x - 5 0

What is the domain?What is the domain?

D: All real numbers except 5

D: All real numbers except -2

x + 2 0f x( ) x

12