kuliah 3-matrek i-transcendental function

7/24/2019 KULIAH 3-MATREK I-Transcendental Function

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MATEMATIKA REKAYASA IMO 141202

Kuliah 3: Transcendentalfunction

ahud ustain


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MATERI Kuliah 3

Transcendental function:

1! Fungsi logaritma

2. Fungsi eksponensial3. Menggambar fungsi

4. Fungsi Inversna


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!e"nition

A transcendental functionis afun#tionthat $oes not satisf apolnomiale%uation &hose#oe'#ientsare themselves roots ofpolnomials( in #ontrast to analgebrai# fun#tion( &hi#h $oes satisfsu#h an e%uation.)*+
http://en.wikipedia.org/wiki/Function_%28mathematics%29http://en.wikipedia.org/wiki/Polynomialhttp://en.wikipedia.org/wiki/Coefficienthttp://en.wikipedia.org/wiki/Algebraic_functionhttp://en.wikipedia.org/wiki/Transcendental_functionhttp://en.wikipedia.org/wiki/Transcendental_functionhttp://en.wikipedia.org/wiki/Algebraic_functionhttp://en.wikipedia.org/wiki/Coefficienthttp://en.wikipedia.org/wiki/Polynomialhttp://en.wikipedia.org/wiki/Function_%28mathematics%29


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In other &or$s( a transcendentalfunctionis a fun#tion that ,trans#en$s,algebrain the sense that it #annot be

e-presse$ in terms of a "nite se%uen#e ofthe algebrai# operations of a$$ition(multipli#ation( an$ root e-tra#tion.

E-amples of trans#en$ental fun#tionsin#lu$e the e-ponential fun#tion( thelogarithm( an$ the trigonometri# fun#tions.
http://en.wiktionary.org/wiki/transcendhttp://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/Exponential_functionhttp://en.wikipedia.org/wiki/Logarithmhttp://en.wikipedia.org/wiki/Trigonometric_functionhttp://en.wikipedia.org/wiki/Trigonometric_functionhttp://en.wikipedia.org/wiki/Logarithmhttp://en.wikipedia.org/wiki/Exponential_functionhttp://en.wikipedia.org/wiki/Algebrahttp://en.wiktionary.org/wiki/transcend


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E"#onential $%o&arithic 'unctions

Dr. Carol A. Marinas


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Table of ontents

E-ponential Fun#tions

/ogarithmi# Fun#tions

onverting bet&een E-ponents an$ /ogarithms

0roperties of /ogarithms

E-ponential an$ /ogarithmi# E%uations


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1eneral FormofE-ponential Fun#tion ( )

* " +here * , 1 !omain All

reals

Range

-5inter#ept6one

5inter#ept7( *8


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1eneral Form of E-ponentialFun#tion ( ) * -" . c/ . d

+here * , 1 cmoves graph

left or right

(opposite way)

dmove graph

up or down

(expected way)

So y=3(x+2)+ 3

moves thegraph units to

the left and !

units up

("# $) to (% #


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Relationships ofE"#onential -( ) *"/ $ %o&arithic -( )

lo&*"/ 'unctions

9 log*- is the

inverse of 9 b" !omain -

Range All Reals

-5inter#ept 7*( 8 5inter#ept 6one

y ' x

Domain All *eals

*ange y + "

x,intercept -oney,intercept ("# $)


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Relationships ofE"#onential -( ) *"/ $ %o&arithic -( )

lo&*"/ 'unctions


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onverting bet&eenE-ponents : /ogarithms

;AER

429 *?

4 is the base. 2 is the e-ponent.*? is the po&er.

As a logarithm(

log3ASE0=>ER9E@0=6E6T

log 4 *? 9 2


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/ogarithmi# Abbreviations

log10- 9 log - 7ommon log8

loge - 9 ln - 76atural log8

e 9 2.*B2B...


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0roperties of /ogarithms

lo&*-M/) lo&*M . lo&*

E- log47*C89 log4C D log43

lo&*-M/) lo&*M lo&*

E- log37C289 log3C log32

lo&*Mr) r lo&*M

E- log*39 3 log*

lo&*-1M/ ) lo&*M51) 1 lo&*M ) lo&*M

log** 7*B8 9 log** B5*9 * log** B 9 log**B


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0roperties of /ogarithms7


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E-amples of /ogarithms


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hange5of5;ase Formula

logam

log*m 9 55555555

logab

log*2 9 log *2

log

OR

log$' ln $ ln


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E-ponential : /ogarithmi#E%uations

If log*m 9 log*n( then m 9 n.

If log62- 9 log67- D 38(then 2- 9 - D 3 an$ - 9 3.

If b

9 bn

( then m 9 n. If C15"9 C52"( then * - 9 2- an$

- 9 *.


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If our variable is in

the e-ponentH.. Isolate the base5e-ponent term.

>rite as a log.


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/ogarithmi# E%uations

Isolate to a single log term. onvert to an e-ponent.


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E"a#les

The follo&ing fun#tions are trans#en$ental

6ote that in parti#ular for 2if &e set # e%ual to e( the

base of the natural logarithm( then &e get that ex

is a trans#en$entalfun#tion.


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7ra#h of a function

In mathemati#s( the &ra#hof a fun#tion fis the #olle#tion of all or$ere$ pairs 7x(f7x88. If the fun#tion input xis a s#alar( the

graph is a t&o5$imensional graph( an$ fora #ontinuous fun#tion is a #urve. If thefun#tion input xis an or$ere$ pair 7x*( x28

of real numbers( the graph is the

#olle#tion of all or$ere$ triples 7x*( x2( f7x*(x288( an$ for a #ontinuous fun#tion is a

surfa#e 7see three5$imensional graph8.


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E l


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E"a#les


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/nverse 0unctions

Definition of /nverse A function g is the inverse of the

function f if f(g(x)) ' x and g(f(x)) ' x.

Domain of f ' *ange of gDomain of g ' *ange of f

1x. Show that the following are inverses of each other.

33

21)(12)( +== xxgxxf

=))(( xgf 12

12

3

3

+xx=


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*eflective 2roperty of /nverse 0unctions

f contains (a# ) iff f,$contains (# a)

3if and only if4

1xistence of an /nverse 0unction

$. A function possesses an inverse iff it is $ % $.

. /f f is strictly monotonicon its entire domain# thenit is $ % $ and hence# possesses an inverse.

-ote strictly monotonic means the function is increasing

or decreasing over its entire domain.


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5et6s loo7 at the following two functions.

a.) f(x) ' x!8 x %$ and .) f(x) ' x!% x 8 $

-ot $ % $

ecause it does

not pass thehori9ontal

line test.

$ % $#

therefore

it has an

inverse.


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0ind the inverse of 32)( = xxf Steps for findingan inverse.

$. solve for x

. exchange x6s

and y6s

!. replace y

with f,$

32 = xy

322 = xy

xy 232

=+x

y=

+2

32

y

x

=

+2

32

)(2

3 12

xfx

=+

Domain of f(x)

,2

3

*ange of f(x)

[ ),0Domain of f ,$(x) ' *ange of f(x)

and

*ange of f,$(x) ' Domain of f(x)


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y ' x

f(x)

f,$(x)


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:he Derivative of an /nverse 0unction

/f f is differentiale on its domain and possesses an inverse

function g# then the derivative of g is given y

))(('

1)('

xgf

xg =

;raphs of inverse functions have reciprocal slopes.


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5et f(x) ' x(for x +") and let f,$(x) ' .x

Show that the slopes of the graphs of f and f,$are reciprocals

at the following points. (# &) and ( )

0ind the derivatives of f and f,$.

At (# &)# the slope of the graph of f is f6() ' &.

At ( )# the slope of the graph of f,$is


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555555555555 25252525

252525252525

2525252525252525252555 555

55555555555

55

In the same way, the inverse of a given functionIn the same way, the inverse of a given function

will undo what the original function did!will undo what the original function did!

"or exam#le, let$s ta%e a loo% at the s&uare"or exam#le, let$s ta%e a loo% at the s&uarefunction' f(x) = xfunction' f(x) = x22

xx f(x)f(x) yy ff(x)(x)

x2

x


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2222

22222222

2222222222222222

In the same way, the inverse of a givenIn the same way, the inverse of a given

function will undo what the originalfunction will undo what the original

function did!function did!

"or exam#le, let$s ta%e a loo% at the s&uare"or exam#le, let$s ta%e a loo% at the s&uare

function' f(x) = xfunction' f(x) = x22

xx f(x)f(x) yy ff(x)(x)

x2

x


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*ra#hically, the x and y values*ra#hically, the x and y values

of a #oint are switched!of a #oint are switched!

he #oint (, -)he #oint (, -)

has an inversehas an inverse

#oint of (-, )#oint of (-, )

./0./0

he #oint (5, 3)he #oint (5, 3)

has an inversehas an inverse

#oint of (3, 5)#oint of (3, 5)


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,$" ,= ,> ,& , & > = $"

,$"

,=

,>

,&

,

&

>

=

$"

*ra#hically, the x and y values of a #oint are switch*ra#hically, the x and y values of a #oint are switch

If the function y = g(x)If the function y = g(x)

contains the #ointscontains the #oints

then its inverse, y = gthen its inverse, y = g(x),(x),

contains the #ointscontains the #oints

xx 00 11 22 33 44

yy 11 22 44 88 1616

xx 11 22 44 88 1616

yy 00 11 22 33 44

1here is there a1here is there a

line ofline of

reflectionreflection


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he gra#h ofhe gra#h of

a functiona function

and itsand its

inverse areinverse are

mirrormirror

images aoutimages aout

the linethe line

y = xy = xy = f(x)y = f(x)

y = fy = f

(x)(x)

y = xy = x


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"ind the inverse of a function '"ind the inverse of a function '

4xam#le '4xam#le ' y = x 2y = x 2

6te# ' 6witch x and y'6te# ' 6witch x and y'x = y 2x = y 2

6te# 2' 6olve for y'6te# 2' 6olve for y' x = 6y 12

x +12 =6y

x +12

6=

y

1

6x + 2 = y


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4xam#le 2'4xam#le 2'

*iven the function '*iven the function ' y = 3xy = 3x22+ 2+ 2 find thefind the

inverse'inverse'

6te# ' 6witch x and y'6te# ' 6witch x and y'x = 3yx = 3y22+ 2+ 2

6te# 2' 6olve for y'6te# 2' 6olve for y' x = 3y2 + 2

x 2 =3y2

x 2

3 =y2

x 2

3= y


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kuliah 3-matrek i-transcendental function

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