t fonyvl po'...1.1 horizontal and vertical translations r2 ( start with the graph: f(x) = x2...
TRANSCRIPT
'Î
Definition
( Transformation: A Cha/JJ!!, IYJa de. -Jv ar. uz__uafton
2 U cJi f hti f +fi e. (J ra p h o f flic f 'u M., fro Il or te! q-Ji·o () al s-o cA a/)~1 <-J ·
.. w,..~. [~XCI Mp,(() Cl re -l/7/M ltdrohS {/ e. sA,'f fs J) (<:. f (e0-17 o /) J J s~~t( icbcs o/ csr« prer.tt (ô ns ·
).\I( .p\\1.l)H on the: on·qi1vd Jf7l/h (or pa/l;.J',+ tJrtlpA) .\? r re_s /;Qr\~ +1r, Nt'1/ po 1' nt·~ on + .. "-~- ~-rZJ l)J ,fo nned 3 rap h . I î\t r·'-:. t~+1 ün s-n t p be.:·h~ttl) 1 .. At. on 31 Acd p9 i "T.s tj, t-~ + rans fonYVL d po' "+·.s , s ctd I e, d ~ 1 !D.ll p ~) 1 ~4- '1
-}In this chapter, we will study the following transformations: • Translations ( 1.1) • Compressions and Stretches (1.2) II Combining translations, compressions, and stretches ( 1. 3) II Reflections (1.2)
These topics cover outcomes R2-R5.
We will also explore inverses, which is outcome R6.
1.1 Horizontal and Vertical Translations R2
(
Start with the graph:
f(x) = x2
Note: This is sometimes called the parent function or the original function.
Sketch: g(x) = x2 - 2 ~ [;) .. 3(~) = ç:r~)-~
( .
L.
---1---1------
h(x) = (x -1)2 h().;J ~ F(x-1
(
~-t------
/~fr-~) The following graph represents the function f ( x) .
Note: This is the graph of the -s1',·ftf+:;).; . I ~I Uhl . it function f(x) = x-2 . '3" Re~ll PC~)~fxl -\Il-
a I -1 a I 1 I 2 I 3 I 4 I 6
I u~, Cl) i t I
' I i L ·1
" " ·- _î_ ~ " i ! I I - .. - - .. - -- -· .. ..
i I ' I I I '
I ·2 I - + - - -· - - .. ·~ -· -- -· "1· -· -· - -· -· I·- -
I I
~- - -t ·· - ·· - ·- I- ·-· " ' ·- -----1 ••• ·-
Sl\.~ch the following translations.
. g(x)'i=f(x+3) ~3 . \ . !J~l
/
(~(x~ f (x) + 2 1' ~ "·· . . \
' -, '· -·-· _ ·--·-· ' ·- --·· ··-- _,,_.,, >, _,,_.__ ··-···· -, ..
-, //
/
/ /
......i---+--+---+--~-+--lt--l'~/--+--+---1-- \/
--+---+-+----+---+----I-+--·-- --+---+- --- -•----+--+---+-- --·
/
' m(x).,/ f(x-1)- 2 .--~ 1 -t~
Effect on the graph Transformation
Example 2: Sketch the graph of y= jx-41 + 3. ( X )j) .-:) ()( t~ lf) ~ +3) ~t.\ A-3" I r y= 1 )(-4 + 3 Y ·~ ,\ l~~----
----l----1-- ""'
X y = bd y= lx -4! y= lx -41+3
_/
Y our Turn: Sketch the graph of y = (x + 5J z - 2. ~ 5 'Ir ;;,__;
pc,n, ba I~ (:.r,J) _,, (x-SJj-;;,.)
--r---- ---- --1---
\. /
/
X
1---+--AI'---+--+- -- - /
-· -- --·l--+---1--'t,-,-\.+.1--,-IL-+'-- --1>--,+----+- ' ,,· --1----1--l--<---1----1-~---+----+--+--+---+---+-
\JJ
Comparing Two Graphs .l ~ 1 1' :;J.. , , UAdc oper rèrn'ori~ ~
If g(x) f (x-1) + 2, sketch the original graph f(x).
0 ', \1, , 2 /',3 ' ... ' .... ' "" '/ '
-- ~o - - ~ I
'<\
g(x) f(x)
-- -- .!. 3 -· L.
-2 --1---
1
I -1 -
0
·,· I
a 1-3 •.-2 , .. '\ 0
I
I
~!~- -- .. -1
- - J - -- 1
I I • r· -
I ·- - - - ·- 1
I ' -•i<r---·--~î··-.
' \, \: ..... - •. I
.3
!
--~-1 ... ~~-~·- I
-3 I
.. - J -4
Note: Ne ~ -full\ 11) +~ 0113 iM I -fu/\t-ho() +hus 1,\1 (', :1 IA(,) d o 11 ft\Q_ (J rl.ft-r/iôl\i ( I, e, ~,-~Iv\ r" ··ffu_ cr pp osî-k orck,)
do
7
y-=f~) Example 3: Determine the equation of the translated function in the formy k f(x h). )"=F(,x-A)+ k
a)
--- - --···6 .....
·- ... _. --7 .
Your Turn:
/
,,,,,,,··
b)
g(x) y f(x)
/
f(x) g(x)
X f(x
/ !-------+-----,
u ~;::::- ---~/-----+-:?, . ·-i·.--· ...
1( ~L __ ,__ -- - .-2 ---- -·-·-1---'-· ---
------- --- -- --~1 I-- --- -------·~- ...
.. -fl .. , -~ -~ .. .. --~ -----·--~---- ·-1 ·- --- ----------- ·-- ·-
1- -t---i--- -·.J.--- -...§ r-- ----
-= ~~==-~t~ =-~I~ =f-- --~-~ -- ~ 4 ~9
(Y -f(~ -4) ~~J • ,....._.,.~,_., •• --. .... ~~,.,...,.,...,...._.,., .............. ~.,,.~~~w•=
q ye'<-) =f(x-4)-~
or o c~) =f(x-~4}+1, ~ J v : -FC x-4) + Cc,
flx ,gx
y=+'(x+~)+-3 (
J'
y ~:. F( ~~Y) v= 9. ·PC)<) .f(X-·1)+3 •• ,# ..... y .... ·- /(x) f ("'"' >~;) Pc~·) y "'""""" y .... , .. .,~·-
\
'~)( y'"···· (J,(X) ·~"';;,
(x·~~·1) y-··· 3 .,~,-- y _ .. ••I• WM 111t,~)t~
- .v y / x)~~ -;.» r.r,~-x ... ~.,,, .... :1.tZ.t:,~ ">! , .. .., Î'\.. y ...
···''' ~·· .. .
- ___ ,,_ -·------------~-·--------
I ;;;lxt \ y-~ .. ~;) I>< I v: f' ) X··· l) +3 •""'~'· y"~"' lxl .. .,,, ..
-
y I ··· x I y ... ,. .......... ,~ ! )< I ""'"'''~ ""'"""~' ·- ._.,... __ --
( ) ~,.,! ?_ ;;,,, y ';). I y ..... ~~,;,il c .,\( 1>J1~1,1t ~· X ''"'Il'" "'~'"' . ~Î..J . ' ( x~-1) + 3 1 y ,,,....~~l = x~
( ) ~ .. y w••"' -)< y ........... ttJl/"-<A'• x;J,.,.
....... ~ .,.,,., .. . '
- ---- ---------------- ... ~~1,.,,,,,. ~ •'"·" ~c-!.):2- 'I ·~"'" ~.;x
fl!'l,};J' I ·x X .. , -1
y,,,_ ~r,1•0.-P•~•,e;,;r,.><n"' ·+ '') 1mlll.' x~·· I l'IJ,JI'
- .\'. I -~ .... 1. y ....... ,r.t~ ... ! y 1 .. 1 .. ~.,,~ ... ,..,u ........
........ ~..- ........ ,.. X . .... )< X ··-
Write equations representing the transformed function after each stage of the transformation. Simplify each equation if necessary.
Horizontal Stretch Vertical Stretch Function and/or Reflection and/or Reflectiµn Translations
.l' =
(
y
y
~--1;;i ·f g f3 c.1 )~ r,d' Lt~+... '°) 1 :,;. b~- - .. , • _ 11, x~CA~,1,1 ~~·-··· ~ ....... It, y~o>i:i s ·-~~,w, ·~·'.r~· , (~
\!:~.:~ ~···~> ~ 't. .,.._.,, ... "~, ..... ,._.. ... ~ .. -.~ ""'/
... St~ rn t ~
978-0-07-073891·-1 Pre-Calculus 12 Student Workbook • MHR 25
(
( '
. :
1.2 Reflections and Stretches/Compressions R2,R5
Definitions
• Reflection = A hl/mt ima5e Ol)1 over or Cl~u+~ 11}\eJ calkd +he I,~ of re t kQ,fion
• Stretch/Compression
A uf-n,l).ff6 t7V)t1r11·on w k,re ·+ht d,:s +r:,nœ fh,M ·+·fui) Ii he of te f (~ r:.+ro n 1 s Mu I -h · p I i' e: d by ot ~c~,I !.,~ fp çio r.
~(l)(t {~t+nrs be.~rwu.-n O ~ i co~ P. rrs si Sxok fac fo,~ 3rea~t·et -/·hot\ 1.. s+rc+cA.
=Jnvarlant Point
A ~~,~-\· +~t)i f~ hot e1/f.rc+c.d by +-Av -imhJ fti('(Y',t:dJOfl (l'.,e. a.i f o ,·11+ fAtif doeJ h (}·1' /Y) 0 re. n?> ff rt: M t/ !f) S I~ fA, sa ltilJ S ~ <J +) .
Reflections -' l .. 2
\
Exl: The graph of Y= f(x) is given.
12
Sketch the following graphs:
Y= -f(x) y=f(-x)
-t ., . -- --j·-
1
.2 •i
i
l ·- ,, .. -- ~- .•. -·· --1 •.
!
2
L -· I I I
.. -- '·~- ·' I
• I ... .,, L I
_! _
l .. _J ••. r.
\ ï - ..... G----fli· 'f = if y.-) ' I
1.3 I 1 I 2 ' 3 1.3 ! I I
I I
I i
I ( .. è
I
13 I
I I - r-- ~ ~- -- - -i - I
... l"
I
-' L -2 ~!,_ . :2. -· ,- - L
I I ·, ... I
I - r
I
-3
I
I I ·, ..
! I .. r ... ·, ·r
I
+ve_ y-va WJ?J' bcco!Ylc -Vt +v~ x-vo luP.s blLOMt -VfJ or -ve_ y-va f u es becoY'f)e. +ve., -v«. X-val.tus bre mt. +ve, ~
Q'( Mt.11-h'ply t/w y-vaUIRJ by -1 tnu 1 ·h' p Jy thL x~vtt lueJ by --1. or ~· ~
r - f( f&z~,f oVlr +Ae x-ct>à s - r( f{a_t;f o\fer the y- a..\"IS ô~ (fllp) (n,';,)
(
lô
y 2/(x) y
Stretches and Compressions 3 ' .. - - ~ -1- . _, - - ·{ -
Exl:
Y f(x) , -2 ,·1 !J '1 ,2 ,3
•.. 1-. I
:.'1 . - ,_ J ...
I -- . -- -· I - - - - - -r -- -·-
Sketch the fallowing graphs:
4
'
Y~ t f(x)
' ' - "· -- - T _5 ... - --, - -- .. ~.
'·4 .·3 ,·1 , 3 , 4 ,5 '.5 ·4 ,-a : 1 ,2 ,3 ,4 ,6
'
)! ~~ f-~~) ' -~ - -1- .,,. .. , ... ~.,., . ' '"' : : : '
: l I'' ,·2 ,·1
. :.1
-2 -2
'.4 ' .. -- .. - - - - - - - ... :. - .. - :4 - .... -- - ., - - ... -:- - .
- .! - ..
' - l, -
' ·5
/'f\~ 111· p iy +N- y-va tws h y ~ =>- V~t· t,ctt I ~ trcfcA l, l e1
Tl"Jc.+ot-- 0 f I/;µ
~ - d 1' V) I dt -t~l\e_ y--v O l«JJ by ~ -> VeI tl'Ct) ( COI'\,, pœJJ'f))I\ by ..
Q fit(:_~ r cj'-f ;;:v l I
v= f(x) I •1
(
~ Ex2:
I
I i :2 I ''" ·- ,J - "' " '' " 1 " " "' "' " "' "" .... ,, "' J ,, ""
I ! I
I 1 '" "· "I " '" " "" "' ·r " '" "' ... " "" '" "" "" "' "l ·• .,, " - , ,- •• " "
I I I I I
0
' - ·• 1··· •· ..
Sketch the following graphs:
y= f(2x) l l , ! • I l _5 I I O \ - - - - - - ~, - - ~ ~ - ~ -· ~ - -- - < - ·- -- - - :· - --- - -- ·1 .. -- - - ,- i .... - - ·1 - --
1
-5 -4 ,-a '
,<f '3 ' \. ' .: \: ' :,
: -, ' ' ' I ' .3 ' I ' I ·4 ' ' ' I
I I
.. ·--··--- .. :------ ~-· .... 6 ··~~- -----·--·--·-· \ ! !
____ ;_, j , 4 _: '·-----l- ' ' --~ . L . -- --,-----~- __ u, T-----,-- -:- -- [ _j_r}--:~·;)
: \ : : : ''.'~tl\'\W " ' . " ' " . " - " : .. ; .. " 2 .. l . " -~ . . ; ... : J ~-! "
I
' ' _ .. ~- _ _ _ ~ _ - - _ 4 ! I ! I : . ·-1 ----ri~;\-- ·:---- T- .. -- _ --~ .. _ . _3 • . ' ·- -· -~:\.- -- - - -: · • • -- · 1- -- - •• - 1 .. -
I "J...;;, ' ' I : 1 I 1
_ ~ _ _ _ 2 .. _ ' -~ ~ _ .... _ L ... _ ..• L _
' ' I '
' ' ' ' ' -·--,---·· r-··-- 1
' - --- - - - - .., - -- - - r· -
' ' ' '
: /: --: ... :""""' ··---~······--,······-1 ... ·;-1·-~-- "
: :, j
2 3 4 I 0 '.5 ' '
' I
I
: ' ' ' ' : ' ' ' I
.,it ,-1 l' l
,4
.3
-6 . I ' .5
I
'' I
'
I g.,
' ' ' ' ' l ' ' ' ' ' '
- tnul-h'C'>' +he. x-va I ue~ - t'Ylultiply tilt X-va~s by ~ ,/I . by Y~ ( cr d 1'v1· ~t x .. -va IUJ.s by &). )J . C hoh'?.D~-h:, f .sk+c.~ by fad?Jr of ~J_, hcn·2or1fr,/ .s+re,fc~ by -Aim, ot ;;J..
- ho~-aih-lr:, f roMptario~ by t,l.c_for of ~ ~
Note: L,.·ke W\'-t~ -b,~c;{)S la-h·ohS) W kt\ ,,.,~ Vii Uu. i'~ assoc(akd - MfA _!.) -flw _oppos~)c ope.rn+n,n +o \-/\v,\+ ,s prr&~l\-kcl I h ~},fi' e_tty (), fi Oh Is pr.:. r ft.)n~ d ,
yq_ r1'·rt1 I .s trcfc~ by '1 At-for o.f -~ Your turn: ())1,fl-f1p \y y-VaU.U~ by ~ The graph of rf(x) is given1b~ (XI~) ~ ( Xi -q )') qf ( )( J r ) Sketch the graph of g(x) = - f(x) and h(x) = f(3x).
4 y l_, rAldi'p ~ ><-value~ by ~
·· ~ ~~·~~·r]~ ~· (x,JJ~(1.\'1 Y) ------ -- 6 __ .... .. ..1~~=~ I I qr· ( ~ 1 )')
y -::: - é I fix) h lJ r·, 2\:>rd·e, f ,J ·h-c'f"c~ by °' ~o t.+nr of ]:
.. X
--1
( y= f(x) l h(x) = f(3x) X g(x) = - f(x) 4
.. ,..~ L1 (-'1) I) (-;;'l, 4) -::;., 0 (-~,a) (-.0/31 a) ô ~ ( 0, 11~) ( ù; ~l) ;, 0 ( ~, o) (~1 o) ~ L\ (C,,J) ( d.1'1)
\3 ,
Ex3: Write the equation of a transformed function
f g(x) ~-~--+--~-2•++---~-L~·~
I I I I
>-----+---~+---~1,-.+----+-~-+------< \ \ \ \ ·----·--·-··· --·---·- ·-·--- 1-2
a)
-12
b)
.. \
-8
\ \ \
-4
y
y
I I I
4 12 X 8
f(x) = [x]
/
/
/
-----+---+---+---+--+--2·+--+---," r---+---+---L--- '<, •
I · -, i -----t--f--,,,t- ---+----t~-3+--f--t-~l------
1 ! '- I H '";·,,_
~" g(x) ,~
( ~ ,J j -;> ( X) 4 )' J y-va~-~ Mul·hplied hy 3
X y = g(x) ,__x __ --i--=-y _= ~g(~x)--i
LY,J) ~ 0<) -y) x-\/~U{JS N\L,d~·,~·(d by -/ ~r
Nt-ud·oh OV·<f· +he )(-OIXI~ X y= f(x) y= g(x) -3 3 -3 -;li ;)) -;)_., -/ I -( 0 () ()
I I --1 ~ 'J - ;;),J ,3 3 -3
r: -+c~) ~ J ck) ~ --.f~) ~· 3 (\) "" - \x I
Your Turn: Given f (x) = x'\ find the equation of the transformation.
g(x) f(x) = xA2
' i/ \ I
4. j--!----1-- I I ; I I I I
- 1-12 --1+--l-+- -
I I I I I I - . -- ----- -18 -- \ I \ I
-----"--1------~1 4 -1~ ---------
" I . I . I \)
~ I
-6 -4 -2 2 6 4
1.3 Combining Transformations R4
Exl: Given the graph of Y= f (x) below. ( \
' 2 ' - -- --, "-,---- ---·- ---,-·-· --1~- -
I I I I I l , I I
__ , ' ----·~1,, i ,_~---
Î I I I I t I !
I I ' î I !
' I '
l •:2 1 •1 , 1 I 2 ! I f i î f !
-- _: '"' ·- ·- _ -· l -- _ _ -~1 __ -· _ .. .J _ _ _ . _:
' -~--- .. -·--·> 1rALIJ*rl)' 1)11-Valufs by -1 Note: This is really the graph 09.,,~x2.~:-~· ~w \tip ly y-values by ~
Sketch the graph of y(-~;: 1~.., r0 ~ r 1. - : :·-~;" __ .: - : _; ~r 3
j I \ ' j I i I
' '
'
' ' • -- ) -. •• ·- - -- ... J - - -- •• - - -- - -- .J .. "
' ' '
( , ·1 , 3 I I ' '
- -- ~- -- .:~ ... - - -- - ~ - -- - - ·- :_ - - ~ - ...
Note: Below are the transformations performed on the original graph.
@. cShl ft 118 ht 1. ~ : ~7"J~+ u gv(r +Ke x.:axi~ @ • Vt~r ·hca l shc·f~1' hy a /ttc,for o~f ;;u
This function can also be represented by the equation:
y :-_- - ~ f ( x---1) + 3 (
I~
' ' ' l I ! -2 ! ! --i--- .. .,f·--·--,-----r···-- ., ,------1····-·· I ! ! l
' ' - -1-.- --- . ·1 - -
' '
Ex2 : Given the graph of y= f (x),
\
·'3 ·2 ;-1
' ., -~
-- .... ~- - . -;3
sketch the fallowing graphs: ,?' 11\Ult- y by :JJ (><,u)-;1 (xJ ~y-1) . ..f'l,'p <)V'c.t· x-~a.\",s . ) I v f":'J (>< )~(X+~ -y
a) Y = 2 f (X) -1 { 1- b) y = -f ( X - 2) . l~· J
' ' ' ' ' ' ' l..). rt1,h+ ,a ' ' • ·····-· ·--- -Ôç-ÔrÔ: -~---·--~ ...... 6 ----:----··-:--···- .. ;·----:---·--·~-- ---·----~--- --'.--------; ----~----6 :·------:-- ~ ·-·--:-- ·····i·-
' 1 l I l I \ I , I
I I l I I I
.. - . - .. ~.- - - .. -~ • ·-. - - ~ .. - .• 4 - - - - - '.- .• .. _I_ - •• - ~ - •• <-. . .. ; . - -· - . . .... i. - . - - L _. - • - •. _4 •.••... ; . - - _ .:._ . .. • l .. . - -:- . _. - ~ - '
- •. - - -- - - -~ - - - - :. - - -1 - - - ~ - -· -3 - - - - ~ · - · - -- - ·· -- 1 -· · - - - ,_ -- - - , - - l I I I
- - -:- - - - .! -·-
' ' ··--·- .; - --1·· - ~-----~---·-3-------:------1-----,--.---:,- --·r"-
' J I ' I l ' ' I I ' ' '
- ~ - , __ ;_ - - •.•• ~ - •• • •• ; ·- - - ~ ·- • - J •• - - ··:· •• ' •••• - ·- _1_, -- -
' : : : : ~ . .. : .. ~ --~~ ~. . . : -~ .. kt)(";>
I ' : /,-', I ' /"'I\"~ : a ' V ; , · ".Z- ~ -+----i--i--t-,---i--,--n--r-~-;+-~x .5 ,·4 ,-3 ,·2 ,., 0 ,..,.,'. , 2 ,
, : ; .r:,:r: --;----·~---·--·~--~~.uM>-; .. ;.1'!,_ ~ --.~~···-;- -•·
' ' ' --~ .. :. __ ... , .... ---~- ----;·----·-~---
'
F •4 ! !' - .. '-, • -- - -, - - ... - - . - ..... - - - --,- - - 1 ... - . ··1· . - ..
I I , : I • I I I
' ' '_ - - ~ __ ~ - O I I --\ --
' ' - - . - - :_ -- : o-:----'l!ll- --~ ----- -·--. --;· ·---·--- ----·-- - '
I , • I
···'--··-·-; ,_.,.,: ~_ ----~ __ :3 ·--- _; ; .,. _
•4 ! I • .. 0 .•. , - - "' - - Ï . "' - -· -,· - - T -
Note: Order is important when combining transformations. » f ci ~ ·~·ve J ~ { ~': f o Ve., r 'X~~ t)x'1'J;' r ,
1 Vitt· idtd :~>h·t ·jt·~.h. b)I t~ .[;,, cAnr O·/ lo l
-.._, ... __ ,,.-, •,c,,.·,. \
6Ve,t y- OJ<13 hori'"l.énf..,I tnuislofiM\ ( 1:- ~) Jwn ?)0n1ZI ( s 1)é fch by q ·f'actnr of îbT
\l i
~ ffip over y~><-,s C) Y = f (2x + 2) d) y = f ( - X) + 1 Up :1.
c~,~) ~ (-><) ~ +,) For (c), it is easier to factor out the 2 before transforming the graph.
Th ~ = + ( ~(x+i) t'ALif'h'p\y x by~) ilf--1· i ererore, y _ . ( x I J ( x, ~ j ~ ;)., - J :J
( \ I
4 ..
( : ; 6
' ' ' ' .,
' :, ' : ; ,4 ! -
' ' ' ' ' ' '
. A ' : / : '\ . ' . ..... '
' ' . '
.5 1•4 , ·2 , .\, 0 / , 1 , 2
: ; ' : : ~- ~- :1··~ - -:- ~--~)f
I ·2 I ; .
·, '
'
' ' ' ' ' ' '
' '
., I , 1 ' ; ' -4 - ' ' ' ' ' ' I _-6 -
' '
-3 =
- : -4
·O
'
' ' •,· '
'
' ( . - \
' ' ' ( e) Describe the order of the transformations that are required to transform the graph of y= f(x) to represent y= -3/(-x + 1) + 4.
Note: Wt co~ :,\"' ~ \,), Work -f 1"1Jt'll. te N +o I\'.) h + a-f'.la -f,i c:-b>n'j ou+ fhe. N.ja-h'vt 111 pml\tks r~ .·. y'° -3 +(-(>r-1)) i ~
1. ~fv.t,+ ovtr x-·Ç\x1·s . . 2. ve, .. kc,( -~+t;;tr-~·-by·~ ~Q:,i·br ot' 3 .: 3. Pa.-P~t-+ ove. r 1-h e., y--01 )(" s ·
4. Htin 2-on+al +mr>J IJrii<ll\ of .1 Uni f -tb +hi!_ Yl'J ~ + 5. v~J~ i')'(O I +·rzm~ )a1i QI) of 4 llhi I~ ,~ r .
p. 3g # 1-·4) <o t-t-1,q) 101 '15 # /~~ C;J.- 1 C.3
(_ i
I~
1.4 Inverse of a Relation
Relation:
R5,R6
In mathematics, it is the connection between the x and y variables. One single x-value may have multiple y-values.
A particular relation where each element in the domain (x) has one and only one (unique) element in the range (y). (Note: vertical line test)
/'\/'v""V~
Note: A function is always a relation but a relation is not always a function.
Function:
Relation but not a function Relation AND Function
----· -- .... +------,-,+--
Sriil'M. }(-Vet lwz for ~ tj., · ffc.Kn+ y~va tues
The inverse of a relation is found when the x and y-values are interchanged.
Therefore, using/mapping notation~imply results in: (x, y) ~ (y, x)
Ex: Write the inverse of the following relation: { (-1,4), (3,5), (6,-2)}
t{41-1)J (S13)J (-;i,G>) 5 Are the two relations above functions? Explain.
C,~0,k (_)\A »v« weJ ! [ (-I, 4) > ( 31 ~ ) ,( 0 1 ~ ~) ~ ~~
~ ( l.1;-· I ) I ( S, 3 ) 1 ( ~ 1 C.,) J ~,,)
Exl: a) Given the relation below, sketch its inverse on the same graph. I I I 1/ i i 3
I - ~ - - 7·j - _, _; -- .. :.. :" ---- __ /_ ~, .. J ... I I /I
! I 1 I I i l ! I !
- ·:- _ _ ~ - I . -· __ 1 . , ' I . -· _ : .. _
! I
I ! ! î I Q !
1 .3 -2 -1 D : 1 2 ' 3
! , ~ /_1 , i ~t,· ()() ,, M.C\~I i f) ~11 /nv, T T; ~ . . : / : i OI dir a~ <mo IM I _: ..... : __ .. : . -2 ····-~r : ,.: .. ~f,,nt17<)f)~
: /; : : : :
(~~ I ) ~ ( \, -;;) ) (
(-Il ~) -·-) ( R1 ~--1) (_\ I ~) -~·~) ( ;;), ) I )
( ~ I ()) -", ( I) j Q)
'/ J , , I i X 1 I ' l I I ...-;.. l l i ~3 i J I \J /i .. - .. - :· - ... - ~ . - .. - .. - -· .... -~ - - ..... : - .. -- r -
r /; î I I j l
Note: The original graph is a function (vertical line test). The graph of the inverse is not a function.
It is possible to determine whether or not the inverse of a relation is a function without having to physically graph it. This can be done by using the ho n''Z.C) '1 +-ct l line test. (
( Gh 1-~ ()f)j i () /1 ( fù i)<2+)~0 I\) Note: The graphs of a relation and its inverse have symmetry over the
line y -=~- '1 Find +k d~Mt/1'1' ,.+ f711\~ of .-jiu. rdtlii QI) 4- Ht in V<.t&G.
b) 'frouvce=fo-~dom·a:tfl&et~l-+mage~~e.--la--fslat.ien-@t-àe-sa - . Fëeipf0€1Ué: - - - * - i'h~(r\ra r. V~. - -~eJ A-otrHôf\ . - -
__.
Domain Range
Relation r ] or [OJ'i] o~ - Q) Q ,p ô~yr~ -2~x-fc9J
Inverse of the r-o,~J OJ r-~, ~] ô, -:::
relation o~ x ~ ,;)._ -Q~ j( ~
When the inverse is a function, we use the following notation:
Y== f (x) ~ Original function y== /-1(x) ~ Inverse function
i I Note· 1- (x) * ( ) . lx
Ex2: If the graph of l (x) contains the point (3, -4), determine a point the must be on the graph of 1-1(x).
(4; 3)
Ex3: If l (x) = 3x-I, determine the equation that represents y= rl(x). )S�,,WJ,, b.,.J- fr) -y ! ( i &t s:-kp J Y - 3><--1 "'\o lis o - s ·k~ rx 4-Y X -z: 3y- I M A
'}] ""~ IS'ola-k 'f . 3 3 I
Y ::::~ httf tvl- Wtrc not ash..d fo sol K lo,{ . 3 Wl WlK as k(__d 1o ~·"d ('{+ / .
. '. f-' <~J ~ ~ 3
QI
d ~·0 °'vVc~ rv Ex4 a) Sketch f(x) {jêi) t,, -1, (/1())
( Û/ I (I, 0)
Will the inverse be a function? Explain.
The, hon·zoninl li Af -krf fz,/{~ ! , f lvz_ /~ vetJc tu, I I /1 o 1- ke a. fo tit f, ·o n .
i ·3 I ·2 I 3 ! ! 1 I I ! ' ' I ! ,; ; J t l
: : i .. ·2 .pt'(~-) I I I I I I
' I ' I ! l ! I . J • .'3 J
l 1 I l l I l l
Can you f. ïc -/-tire t:JAlY1 11 ~!I )od k. l, k ? · '_; V(f-t,r41 J,,_'tu_ .f~.r J <if ( rr. / (e ~ ·-f ove r (J ru. LJ ~: ~"Y , I 1,1 ou GM t) q_~:} J /\ rage f111 fa li h. we,;
. V . l.) . . p t( J, t /c d ~ /) 0 ,f IV b) Use the graph of f(x) to sketch the inverse, .r-1(x). ft1flC 1i"ôf\
If f (x) is the inverse of g(x), then f (g(x)) = x ,
Proving functions are inverses of each other
It is difficult to prove that two functions are inverses of each other graphically. To prove that two functions are inverses of each other, we must show this algebraically.
Essentially, what we are doing is substituting x into g(x) . We then substitute this result into f ( x) .
If we end up right back where we started, then we will have shown that these two functions are inverses of each other. If we do not end up where we started, then the functions are not each other's inverse.
( f(x) g(x)
Ex: Show that /(x) = 3~;;@are inverses of each other.
f(jf__~l) = 3 ( ) -s ){f5
.-.i-...w,.._
3
f(X¥) z: Xf 5- 5
·- X p ( J ( ~··} ) ~ X /,
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Practice: Show that the following pair of functions are inverses of each other.
( a) f(x) = x - 4 and g(x) = x + 4 f<j[~J)=( )-4
x+ '' f ( ><r4) =· (>r·f 4) - 4
= X , '. f(j (->r J J =- x· i'. i" verte.~ v
x-3 b) f(x) = 8x + 3 and g(x) = -8-
f ( j~J ) -= R ( ) f 3 x-J ...... ....--"'11'1
f (' :-3) ~ l~(r.~ J t3 =r .~ s: x -3 f3 :::: X
, 1, f { J c ~ l ) = x .: /,, v e: res i/ x-2
c) /(x) = 2
and g(x) = 2x + 2
. ~ J~<~) ~ ~··· L{- ~. . ;;)~Xi" ;t,
f (J.k' +~)-) = ( ~:_!?J~.:il ~ - .:Lx tl-~
~..,lJr-,.<1,.W,...,,,~.MIM>
~ - ~,.,}(' - ~
=,,.. ')(
.. f(JC+}) ~ x .: /1) ~r~J
(
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