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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 $oes satisfsu#h an e%uation.)*+
http://en.wikipedia.org/wiki/Function_%28mathematics%29http://en.wikipedia.org/wiki/Polynomialhttp://en.wikipedia.org/wiki/Coefficienthttp://en.wikipedia.org/wiki/Algebraic_functionhttp://en.wikipedia.org/wiki/Transcendental_functionhttp://en.wikipedia.org/wiki/Transcendental_functionhttp://en.wikipedia.org/wiki/Algebraic_functionhttp://en.wikipedia.org/wiki/Coefficienthttp://en.wikipedia.org/wiki/Polynomialhttp://en.wikipedia.org/wiki/Function_%28mathematics%297/24/2019 KULIAH 3-MATREK I-Transcendental Function
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In other &or$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/transcend7/24/2019 KULIAH 3-MATREK I-Transcendental Function
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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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