-'c0. l-'X-t').) = c . +i> G-·C:) ::

Graphs and Analysis of Logarithmic Functions

Analytical, Graphical, and Numerical Approaches

The inverse of an exponential function is a logarithmic function. Let's remember some properties of functions that are "invertible." First, functions must be one-to-one functions in order for their inverse to

· exist. Summarize how to determine if a function is a one-to-one function or not. NvVl'\�'� J IA. �� '� \-\ �'vf r'\O �-IJ� 4�al.

�� , � � 0-- I -1 i:� "t- p�� hl)(-;2fv'.W llvUt �t.Consider for a moment the exponential functionf(x} = -2x+3 + 2, which is graphed to the right. A table of values has been provided to remind you of the ,······--:--······,········;"·······,········;"·······;"··!} ······v·················v·················v·················,

transfonnation of points from the graph of y = 2x. l:::::::r:::::r:::::r:::::i:::::::r::::r: ...... l ....... .L ..... 1 ...... ..1. ...... 1 ..... ..l. ......lPoints on the graph

ofy =2x

(-2,1) (-1,1) (o, 1) (1, 2)(2, 4)(3, 8)

1 -4 1-, 2

(-3, 1)

(-2, o)

(-1, -2)

(o, -6)

Transformed by (x-3, -y+2)

(-s, 1i) (-4, 11) (-3, 1)

(-2, o) (-1, -2) (o, -6)

1 l i i i i i i 1 l : : l :

i i i i i i i 3 \ \ : : : \ ; r--: --·--4---+---�--r- _J __ J __ ,l. __ L __ L __ l __ J

!·······-!---·····i······+·····:·······!········i···l· ······!,········!,········1,········

!,········1,········!,········i.

: : : : : l :

. . . . . -;7 45 -5 --4 -:3 -'2 -'l i ········:---····· i ········:---····· i ········:---· .. ;-l· i i i 4 i j i

l 1 j i i 1 1 � ........ j ....... ) ..... ) ...... ) ........ 1 ....... 1 1· ...... i ........ i ........ i ........ ; ........ i ........ ; ........ �

l l l l l l : . �-·······!········?········!········+········!········? ...... L. .... .l ........ 1 ....... .1 ........ 1 ....... .l ....... .l ! .... ) ........ 1 ..... ...1-....... ! .... ....! ........ ! 5 ...... ! ........ ! ........ ! ..... ) ....... ! ..... ...!. ....... ! : : : : : : : � ........ � ........ f ........ j ........ f ........ j ........ f ... 5 ...... l.. ..... .L ...... L ...... L ...... l. ....... L .... ..l i i l 1 l i l

:······+······:······+······:········:········: 7' ........ ' ........ ' ....... .L. ...... L. .... .' ........ ' .g ...... ! ........ ! ........ I ..... ...1 ........ 1 ....... .1 ........ I

-;7 ... + ... :5 ...... ; ........ ;.Js--� ""'.1 1-; 1 1 l i l : !········l········!········!········!········i········!-· : : : : : : : � ........ � ........ i ........ � ........ � ........ � ........ ;3. � ........ � ........ i ........ , ........ i ........ [ ........ +-4 ! ..... ) ....... ! .... ) ........ ! ..... ) ........ !-s· i l 1 1 � l 1 r········1········t· .. ··· .. 1··· .. ···t··· .. ···1···· .. ··t-5 :········:········l········i········:········:········l7 ' ....... .: ........ L ..... .: ........ ' ........ ' ........ '-11

i j 4 s j i ...... L. ..... ,

L ....... L. .... .L ..... L. ..... 1 ........ 1 ..... :, ...... . "······:·"····+·······�--..... ,: ....... ,:

:;j::r 1 l:r 1 ...... : ...... .J.-...... : ....... .:... ..... 1 ..... ...:. ....... :

Now, let's find the equation of the inverse ofj{x). Remember, t analytically' nd the inverse, what do we do to the equation?

0 SuJ·A-� � � � \\I\.,� �� @) So\� � J\.L�ult� ��an -for .j, �

�kc_e_ 1 v-> ,� �-'c0. �� Perfo� the proces� you just described to the exponential function�to find the equation of the mverse functi�1,f1(x).

'f-=: - � 1 -\- � �� l-'X-t').) = y,- 3

?c -:::: -d- y·-t 5 � ::v '1+3

·-x..-i-= - ;l ·-'X_.-t� -= � y+-�

-1-0,g:,. tx.-+�-:: � 9.. l't

-+3

" = - 3 ·"" �2. c_.�+i>

G-·C:) :: <3 -t 1oj,. L--x-t.?)j

For each of the functions below, fmd the equation of the inverse function.

1. f(x) = 2-x+3 -3 2. f(x)=3+ex+23. f(x)=-3x-2

+1

)( -;::. - b -r \

� --:>� -\-:._ -3' y-?-.

-�-t \ =- 3,

t&�?, (-'JG+,) -:.. '-J -')__

'f ':=.

2- -+ J-Oj /.-')(.-+ 0

['.&) ;: Q_ -t !,o� (_.--x_ -H)]

On the prt. ..is page, you found equations that represent inverse functior. three exponential functions. Complete the table below of i .nation about the exponential functions and their inverse logarithmic functions.

Exponential Function f{x)

f(x) = 2-x+3 -3

f(x)=3+ex+2

f(x) =-3x-2 + 1

Domain, Range, and Horizontal

Asymptote off{x)

D: C -Ol)) ex:>)

�-- (-3J co)

Domain, Range, and Vertical A�of

ci-�D� (-3.1 �)

Equation of Inverse Function /-l (x)

R ". (-(X)-> co) It-'&)-=- 3 - Las� (x�3)

l1Pt- � 'f = -3 I '\I f\ •. ,x -=- -

D� (-co.;�

R:, ( 3; a:>)

\-{ Pt �o j -=:_ 3

D�(�co)

� � (-d).) 0 \-\Pt� �-= \

1) � (�� Q))

R � (-oo} CD)Lr'c,.) -:. -l \- �l:x - 3) I

\Jc-\·� 1' �3

1) ".. �00; 'i') 1

'R '. c-�., o:>) 1-r-'&. ')-: � \- �1 (-x �tI

\l'Pt', � :: l

Set the argument of /-l (x) = 0 and solve

forx.

)( -;- t> -:::. C

')( ::: -3

x- 3 -=-o

)( :=. .3

-·x.-+\-:=:..O

-")'..,;:::: -\

'X.-:::.

Set the argument of J-\x) > 0 and

solve for x

�-t-3 >O

. 'X, '>-3 (-3� cq)

%-3 '>()'X., > 3

(3 ., 00)

--X,-t\ >O

-x. >-\'X, L. \

t-=a,) 0

Based on the infonnation in the table on the previous page, what inferences can you make regarding how to analytically find the following characteristics of a logarithmic function given the equation of the function?

How to Find the Equation of the Vertical Asymptote of a

Logarithmic Function

How to Determine the Domain of a Logarithmic

Function

How to Detennine the Range of a Logarithmic Function

Find the indicated properties of each of the following logarithmic functions.

1. F(x)=-2log2(2-x)

Equation of Vertical Asymptote

-�- % -::?. 0

Domain

Range

-')(_,-:=:-�

5,:: i1

�-')G>O

-� 7-�

� L...�

(-oo; ::l

2. H(x) = -3 + ln(2x - 3)

Equation of Vertical Asymptote

J.....y.. -3 -::- 0

Domain

Range

�)(� 3 -�x� 3/�\

�)(-6 > O

��73 X > o/;i...

Does the graph lie to the left or to the right Does the graph lie to the left or to the right of the vertical asymptote? , of the vertical asymptote?

� a,h R'0 u.� � � ,4... K�) �� 4-0 � i � <t:> X :;.� 1:,\c_ -\-o � �'\- ct x�.Y,� �6b �) \� \,\L � ' � \1(�-oo � . \b 3,6. Q:) ..

The table pictured below is a table of values representing an exponential functio 'f(x) = ex-2 -3 Usethe table of values to answer the following questions.

X -5 -2 0 2 4 6

j(x) -2.999 �2.982 -2.865 -2 4.389 51.598

a. What is the equation of the horizontal asymptote of the function,.ftx). Explain how you know.

� � -� \.S -\-L.c.. �'� °'-��-to, �\c:_

� rx., '"7 _ [)); � W � -3 -b. Is the graph of.ftx) above or below the horizontal asymptote? When the inverse function,f- 1 (x), is

graphed, will the gri

h be to the right or left of the vertical asymptote? Explain your reasoning.

� � �J �� oJoe,-v..e �� ,�W �&N�e \e,\c._ CLO.l

"'1'� .o:h 6-) >-�. � � 1) ,-f--\(�) w,li l>Q. +o � JW� 4) �-\-t; vu-�c.,J Cl_S�e �le... � v�

ci � ';>-3 � -\1-l&-) ,I

c. WM.at is the equation of the vertical asymptote off-1 (x)? Explain your reasoning. �-=- -� ,� � ..J'�.-h'ul.. °'-��'1'>-fe d... �-'6c.) \otc.. j=-3

,� � �".t� OL<:::.�\v'l-e' -P -F.&Y. d. Identify the domain and ra� of the function�(x) :.lnd its inve�s�, J-1 (x).

� -: ( -cl)) cb)� � (-s1 ou)

e. Find the equation of the inverse function,f-1 (x). Show your wor

� = e 't ·-� - 3 ! ..........

r-------

r--------

r--------

-r .......

T---------

r----

:

rx�� e 'i-� 11 i--\ l-!-1: _k. e 'i -� j�:j l :[ t��!:.�[�,� l-x-\-� =

�c.�+2>) - 'J·-�

f. On the grid to the right, sketch a graph of J- 1 (x).

l-----------l--·-.... ---:--......... 1 .. -.... --+-·-----i-·-·-·-·-+ ....l.: : : : : : :

,7 � f -l4 -1

\ \ . .!...... ! . l-l l- : [_ __________ [. .......... i .. ---.. --.J .. , ____ J -------! ______ ..... ! ... :j ___________ j ________ J _______ J .. ___ ..... ' __________ j ________ ..J. ___ 5

� � � 1 I 1 = ........... = ........... = ........... = ........... = ........... = ........... = ... 7

i ! i i tJ.i \ l l l 111

-

'

The table pictured below is a table of values representing an exponential function, f(x) = -2x+2 + 3. Use the table of values to answer the following questions.

-1 2 4 7

1 -13 -61 -509

a. What is the equation of the horizontal asymptote of the function,.f(x). Explain how you know.

d-=-- � \ � � �'� A_btt\i:A-e_ b\c_ o.D·'X-� -co.) �c�) � �#

b. Is the graph of.f(x) above or below the horizontal asymptote? When the inverse function,J-1 (x), isgraphed, will the graph be to the right or left of the vertical asymptote? Explain your reasoning. � � 4> �(�) \ '> 'o� -\{J_ V\b(l� ct&�� lo\c.cnll \J'� 'l) �L..3. � u+- � -f-l(x_� \l&.,l.)�Lllle, 4{) � � '1J \ .\-z, -J e.-rl,' uJ <>-.S o"i'�,k (,,le "'1.L U�d "X, � � \.,V'"'\ f- -\ c..�) .

c. W\ltt is ·the equat10n of the vertical asymptote off- 1 (x)? Explain �our reasoning. � � 3 ,� -\"L..e.. ut.w--A""'� Cl. s� -bt-.e b\c � =- i Ls� \A.e<'t � �bM+o� "1) +6c-).

d. Identify the domain and range of the function.f(x) and its inverse, J-1 (x).

f&-) --::> �: (-a::>.}ex>) �: -t-'(�) -,, �: (-a,.,> � � ';,

(_-cOJ 3)(-ooJ w)

e. Find the equation of the inverse function,f-1 (x). Show your work.

X -= - � '-/·n. -t- 31 ·1·rr·1-r·1·: ::1 :r :r:, -1 �I :i

� - '2_ :: - '"' '-I + 2- : ........... j .... ...... j .... ...... . : .... ..... + ......... : .......... ,:.. ... s. .·.·.·.·.·.·.·.·.·1,.! .

. . ·.·.·.· .·. · .·.· . · . · 1,.! ... ·.· ....... ·. ·. ·.·.· !,1.�· .·.·.·.·. ·.·.·. ·.·1 ,.

! ... ·.·.·.·.·.·.·.·.·.·1,.! .

.. ·.·.·.·. ·.·.·.·.·.·1,.! .

.. ·.················· ·::! J O' 1

............ 1 .........

) ......... ...1 ......... ...1 ......... ...1 ......... ...1 ...... 4 !I

1 1 I = l � l ·- ,r'\/ .J.. 3 -- r) 'I-\-� ! ........... ! ........... ! ........... ! ......... .. ! ........ ,..! ......... ..! ..... 3. ' ' ' ' : :

,._ , o,'.... , , , , , , , : i l i i i i

Jio� :2 L -'ll -1-3) -=- 1.os" :1 'I + :L l--1---4-1-4-·i_ ......... .J..-!-4-!-+�J_:::::::::�! :,-I..... -+i--1--! -1-! -+i--1--! --1--! ___.i 0

I"\ �7 �-li-fl�-{l� � 4; 1 �· ·� 1 l\ "'3 II I -rx--+ -=- '1 + d- ; ........... :,.. ....... }""""}""" ... i .......... ; .......... }·-!· : : : : ' : ...)V'J .,....'--= I i 1 1 1 ; 1 � : = = : = : =

: ........... : ........... : ........... : ........... : .......... : ........... : ... 2 ......... ! ..... .... 1 ........... , .......... 1 ........... : ........... 1 ........... i ......... � ........... : ········ 1 ........... ) ........... 1 ........... ] ........... )

-I i +-+-i--i -I·········:···········:····· ····:···"'"'"':'·'·'*"''';'''""""''':·······'"'': ......... 1 ........... 1 ....... l ....... ,:.. ......... l ........... l ........... l

' ........... ' ........... ' ........... ' ........... ' ........... ' ........... ' ... v ......... ' ........... ' ..... ":/ ........... ' ........... ' ........... ' ........... '

f. On the grid to the right, sketch a graph ofJ-1 (x).

Name �S� ¥":>

Period __ _ Date

Day #38 Homework

Complete the table for each of the exponential functions below. Be sure to give justification when asked to do so.

Function

1. F(x) = 2x -3

2.

G(x) = -(l \x+3

21 -1

.) .

H(x) = (l.25)-x+l +3

4.

f(x) = (1tx-l + 2

What are the domain, range, and horizontal

asymptote of the ex_Qonential function?

'[): (-a:,, c:,o) R: (o ) co)

l-\Pr: � -: 0o: l-ooJ oo)

R: (.-001- i)

HA--: �r= - \

1): (.- (X).) ex>) R: (�,QO) HA--·.�-:. 3

0: (.-�co) R: (", oO) HA-"· j-: �

Is the graph of the function above or below the

horizontal asymptote? Wh_y?

S,\'\c..t.. o..>0 > �c.,..) \ C, 0,.\)0� � -:. 0

5 \¥'C...< o....r:::. 0 I

G.,.t,..)

\ C, \rJe..\� j= _,

What are the domain, range, and the equation of the vertical asymptote of

the inverse function? D: (o ., oo) R ·. ( -ex>) QC))\IA: x:.o'O: (-co., -i)

R: (-o0J ex:,)

"r..r·. ')(:-\ S,� °"->-o, H(}(.) 11': (-;., �) i� �'v:>o� �;;. IR·. (-ooJ oo)

'/{+: X '=- .3 S,""CA.. C).. � o) ,- �) 11': ( 'l.., co) i '=> (). \:io\).f �-: " I R : (.-oo., oo)

\/Pt: 'I..::�

Is the graph of the inverse function to the left or right of

the vertical asymptote? Why?

S\""'c.c. F(�) wo..s IO..\,tN,(. � :OJ � �-'(,c.) \ s. 1°C) �('"ict� d{ X :o.

� V

�\\f'\CA. (,..U') wo.o �"'-0 '1 ::-\, � s--'(>c.)

\� -\,, -\k � ' X =-\.

S,"'<..t.. \-\�) �

c)..� �=3., � \-\-'�) \ � "ti) � \""\C\ \.vT <\ 'X.-:. 3 . --

S,V\C,A. i c,.) vJO.S (l..t&JQ_ �-::2., � T-'�) \� 1D �� -=a.

For each of the functions below, find the equation of the inverse function. Show your work.

5. F(x) = 2x- 3

ll - � x� ��'{-�

� 2. ',( : �:a.�

�2.�-= 'j-3 3-t�,.

x ��

6. G(x) = 2ex-2 + 4

'X = � e..!:r 2.. "'t' �

)(.- 4' � � e'1--:l.

For each of the logaritlunic functions below, state the equation of the vetiical asymptote. Also, state the domain and range. Show your work.

8. G(x)=log2(3-2x)+2 9. F(x) = -2 + ln(-x -3) 10. H(x)=-log3(!x-3)

\JPr: 3-�x-=-O \JA-: -'X -!>� 0 \fA ·. �x -!>-= o

-;l.)(.� ... 3� -'X., � � �x� 3 tx � ,;,;\ fx=-3) (><� �

n: '3-:l.x. >0 p·. -x-3>0 1) ·. �)(.-3 '>0 - ;2. )(. ';) -3 -x>3 ..Lx >;

x� 3/:l. X C:::... -3a-

x.>� \�-=) �h)j LL-co.)-��

t<<-J oc1 R ·. \(- oo, co)\ R: l-coJ oo)

R: 1C. -a,) �i

Consider the function g(x) = -(2 y-3 - 2 to answer questions 11 - 15.

11. Explain how the graph of g(x) = -(2 y-3 - 2 is different from the graph of.f{x) = (2Y.CD ��t.kcl � -x. - C?l)4 er..

�� . �3 ��6'\;-ecl � �

(X-t3, -� -�)12. Based on the transf01111ations you described in exercise 11, complete the following table of values.

X

-2

-1012

13. Find the equation of g- 1 (x).

)<-: -�y-'S_�I'\ - '\ 'f -� )(-\- ... ::. .,...

-x-'l.-:. d--y-3

f(x)

'(-� �� (-x -:i.J :: �:i �

��f)(-1) � y- 3

'J -:. � -t �,.. (-x-'.Y

� -1ei,.') '::. ?, .... �-:.. (.--x-;).)� --=

15. Domain of g- 1 (x): ( -00) -�"")

Range of g- 1 (x): (-00)

fl:>)

CoordinatePoints of g(x)

Coordinate Points of g- 1 (x)

( \

'5

14. Sketch the graph of g- 1 (x).

1-11 -1111 l t :+·········,···········,l::::::::::r:::::::r: ....... i .. ::::::::::::::::::···i-·::::::::r:::::::r::::::L::::+·········,··········

11 i \:i \:i-l \: !-········+·········!········+········+········+·······+···· ·'··········l······l·-+-································ 1 : 1 I l I

� � � � � � � � � i : : : i. : : :, i l 1 l l : : i······· ···i· ·········i···········:······ ····i···········i···········i········l·······-j··· l

i ........... i .......... i .......... .i .......... .i ......... .i .......... .i ........ . \ ......... .i ... :

......... · ......... � .......... 1 : . :

; ; ; ......... · ........... · ........... ·

Below is a table of values for the exponential function/(x) = ex-2 -3 . Use the equation ofj(x) and thetable of values to answer the questions that follow.

X -5 -2 0 2 4 6

J(x) -2.999 -2.982 -2.865 -2 4.389 51.598

16. Fill in the table below identifying the domain, range, and asymptotes of the graphs ofj(x) and/-1(x).j(x) 1-'(x)

Domain (-00,00) Domain (.-�J co) Range (-3,o0)

Range (-Cc), co') Horizontal

�"a-; Vertical

'X. :-3Asymptote Asymptote

17. Is the graph of/- 1(x) to the right or left of the ve1iical asymptote that you identified above? Givea reason for your answer.�\"'� o.il... "� � �()')4'.-3 l � 3� °'b fl�)\� 'oe.\w.,L--k "'°c-·, 3°"'� �N� � �-: -3. � � � � 9, � -•c ... ) "";LL ...,� *° � � ei> it<; "��cJ. ��.

18. Find the equation off-'(x). Then, use the equation to find the equation of the vertical asymptote.

'I.': e.. y-"2. - 3

')(. -t- � -: C-'I- ':J..

�l')(.-t� ":. .k e.'l -'1

� lx-t-�J � y - �

'I =

19. Sketch a graph ofJ-'(x) on the grid to the right.

1 ........... , ........ 1 ........ , ........... 1 ...... :.+ ......... , ........... , ........... , .......... , ........... ........... ,, .......... .

!··········-j--·······l-······!···········!····-5·

1 r r r-r·-i-. ..... ...:. ................... J_J..�=r"

1:::::::::1::::::::r :··::.i ... :::::::f :::::+·········!···········,···········!···········!···········!· .......... ; ......... .1 I I � l

--5 -4 -ii -h

! !-+ [ i :II ii-I�

i ........... i ........... i ........... i ........... i .. 5.

-'c0. l-'X-t').) = c . +i> G-·C:) ::

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Transcript of -'c0. l-'X-t').) = c . +i> G-·C:) ::