37
FUNCTIONS & GRAPHS FUNCTIONS & GRAPHS FUNCTIONS & NOTATION FUNCTIONS & NOTATION WEEK 10 WEEK 10

FUNCTIONS & GRAPHS · 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

  • Upload
    others

  • View
    1

  • Download
    0

Embed Size (px)

Citation preview

Page 1: FUNCTIONS & GRAPHS · 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

FUNCTIONS & GRAPHSFUNCTIONS & GRAPHS

FUNCTIONS & NOTATIONFUNCTIONS & NOTATIONWEEK 10WEEK 10

Page 2: FUNCTIONS & GRAPHS · 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

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)

Page 3: FUNCTIONS & GRAPHS · 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

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

Page 4: FUNCTIONS & GRAPHS · 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

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

Page 5: FUNCTIONS & GRAPHS · 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

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..

Page 6: FUNCTIONS & GRAPHS · 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

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

Page 7: FUNCTIONS & GRAPHS · 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

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}

Page 8: FUNCTIONS & GRAPHS · 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

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

Page 9: FUNCTIONS & GRAPHS · 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

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

Page 10: FUNCTIONS & GRAPHS · 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

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)}

Page 11: FUNCTIONS & GRAPHS · 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

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

Page 12: FUNCTIONS & GRAPHS · 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

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

Page 13: FUNCTIONS & GRAPHS · 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

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

Page 14: FUNCTIONS & GRAPHS · 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

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

Page 15: FUNCTIONS & GRAPHS · 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

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)}

Page 16: FUNCTIONS & GRAPHS · 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

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.

Page 17: FUNCTIONS & GRAPHS · 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

(-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.

Page 18: FUNCTIONS & GRAPHS · 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

(-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

Page 19: FUNCTIONS & GRAPHS · 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

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.

Page 20: FUNCTIONS & GRAPHS · 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

#1 Function?

Page 21: FUNCTIONS & GRAPHS · 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

Function?#2

Page 22: FUNCTIONS & GRAPHS · 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

Function?#3

Page 23: FUNCTIONS & GRAPHS · 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

Function?#4

Page 24: FUNCTIONS & GRAPHS · 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

Function?Function?#5

Page 25: FUNCTIONS & GRAPHS · 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

#6 Function?

Page 26: FUNCTIONS & GRAPHS · 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

Function?#7

Page 27: FUNCTIONS & GRAPHS · 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

Function?#8

Page 28: FUNCTIONS & GRAPHS · 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

#9 Function?

Page 29: FUNCTIONS & GRAPHS · 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

“f of x”

Input = x

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

f: x yf (x)

Page 30: FUNCTIONS & GRAPHS · 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

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))

Page 31: FUNCTIONS & GRAPHS · 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

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

Page 32: FUNCTIONS & GRAPHS · 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

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).

Page 33: FUNCTIONS & GRAPHS · 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

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

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

f(0) = 1

f(-1) = 4

f(2a) = 12a2 + 1

Page 34: FUNCTIONS & GRAPHS · 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

f(x) = 2x2 – 3

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

Further ExamplesFurther Examples

Page 35: FUNCTIONS & GRAPHS · 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

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

Page 36: FUNCTIONS & GRAPHS · 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

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

Page 37: FUNCTIONS & GRAPHS · 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

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