ࡱ> RdO)_zpPicturesPowerPoint Document(ASummaryInformation((2  !"#$%&')*+,-./01456789:DocumentSummaryInformation83Current User$nVvE/i-!7)PNG  IHDRǮgAMAPLTEic]VPJC=60*#  &,39?FLRY_flryyrlf_YRLF?93,&  &,39?FLRY_flryyrlf_YRLF?93,&  &,39?FLRY_flryyrppp```PPP@@@000  pHYsN7IDATxr0 D9G_&lp;ъ3mם9&N ,<*wsR9sOiܜsvg֔uSbKWsڕG&P0'T=;|/T=;&'n|*MOin$ӴYO^VZr8jSi~[BIph>yNUO^_cQ#|+iEp$מ&ӂO  PuZ˧7~7Z>MBOӷv<K=9ߟ*Z>mO:A *Rn49Ooo0N? ·4C=կ>|Qi'}T>yNKˍṅ.>XZpt pDٞ\K+"J:}T>y"ty"b]ƐO 9=_u>tGؠ>/hvj6O lND\BS :5O%>h39U=fϰ6 zyl^s~ܟbK^W9ΩؠrÖס\>EuN۔ϧ,\9 .( N/0(  0;[0 0 000$([\{b00 000000000  0=] 0 0 0000 2 3 !A0C0E0G0I0c00000000000000000!%),.:;?]}acdeghijklmnopDTimesNew Romap0MpIxpIkDTimes New RomanpIxpIk@  @@``   @n?" dd@  @@`` |tH+   ,b$VvE/i-!7)H `1?@i==@ g4kdkd4dppp@ <4BdBdg0Mp Jp? % 6.3 Inverse Functions&2001 by R. Villar All Rights Reserved'' Inverse Functions An inverse of a relation (set of ordered pairs) is obtained by switching the x and y in the ordered pairs. For example, the inverse of {(0,  3), (2, 1), (6, 3)} is: {( 3, 0), (1, 2), (3, 6)} The graph of the inverse is the reflection of the graph of the original relation. The mirror of the reflection is the line y = x.I=a==i  =QibExample: What is the inverse of y = 3x  2 ? J2  = = Switch the x and the y. The inverse is x = 3y  2 Inverses of functions are found the same way& however, the inverse of a function may or may not be a function... For example: Find the inverse of f(x) = x2. Is the inverse a function? Replace f(x) with y y = x2 The inverse is x = y2 Graph each by making a table of values... Np io!==@==ii PiPi  BOriginal function: f(x) = x2 Inverse: x = y2C== ii0=   PComposition can be used to verify if two functions are inverses of each other...$QP<hIf the functions f and g are inverses of each other, then f(g(x)) = x and g(f(x)) = x Example: Verify that the functions below are inverses f(x) = 2x  1 g(x) = 1/2 x + 1/2 f(g(x)) = f(1/2 x + 1/2) = 2(1/2 x + 1/2)  1 = x + 1  1 = x g(f(x)) = g(2x  1) = 1/2( 2x  1) + 1/2 = x  1/2 + 1/2 = x Therefore, f and g are inverses of each other. :; == =6i i=  i i i"= ii=%= i,==Pd ` ̙33` ` ff3333f` 333MMM` f` f` 3>?" dd@,|? " dd@   " @ `"  n?" dd@   @@``PR    @ ` ` p>> |( 䡤ܡ,     `wawa1 ?P   T Click to edit Master title style! !@  Zwawa1 ?   RClick to edit Master text styles Second level Third level Fourth level Fifth level!     SB  s *޽h ? a( &Microsoft Office 98 ZR@( ( A @  Z$wawa1 ? @   RClick to edit Master text styles Second level Third level Fourth level Fifth level!     Sp  01 ?    B  s *޽h ? a( P 0( aP"R%""  B  s *޽h ? a(d H$( wY Hr H S ` @0   r H S d `     H H 0޽h ? a(x  Ȱ   0 (    # lwawa1 ?0Pp     # lwawa1 ? <$   X  C AGrid 2 000028FD Macintosh HD ABA78158:U @,$D 2  6=1?p  ,$D 2  6=1? ,$D 2  6=1? ,$D B  HDo? p ,$D 2   6i1? P ,$D  2   6i1?0 p ` ,$D  2   6i1? ` ,$D     H1?i 6c ,$D  Ioriginal relation=   H1?@ Z : ,$D   Qinverse.i=  H$1?``Z,$D  V mirror y = xH  0޽h ? a( ` \T`(    # ltwawa1 ?     # lwawa1 ?`0 <$   H  0޽h ? a(  < A 9  4 (   4 4 # lwawa1 ?    4 # lwawa1 ?p0P   p 4 HA1?0 2 4 TZG=p?h  ,$D @2 4 C C_TENGHK I`TJ_TQ=p? _TLT_TLT`T_T_TLT`T_T]  ,$D 2 4 TGip? . b/ ,$D  :2 4 3 C_TENGHK I`TJ_TQip? _TLT_TLT`T_T_TLT`T_T' U ,$D  B  4 <Ԕ?@P,$D    4 Zwawa1? ,$D  by = x2@==    4 T$wawa1? 0~E ,$D  Tx = y22 2ii  4 Twawa1? `,$D  My = x, 2,f 4 Twawa1?` /,$D   LThe inverse is not a function since it does not pass the vertical line test?FM 2i/H 4 0޽h ?O@4444 a( , \T<( dӾ8 < < # lwawa1 ?`p    < # lwawa1 ?`0 <$   H < 0޽h ? a(r4`  6#/ 1>! ?MAMg. .Word cannot create the mail merge document.Do you want to replace the existi Oh+'0t  8 D P \hpx'Warm UparmTechnology ResourcesrPechechechSCHSolo15SMicrosoft PowerPoint 4.0oi@2@@d@@ӳ5!GPICT c@@2pd 2HH #&42#{~x,"!xh`#C#rxxY$A``F#``)MT*=$``` c C.,d x$x$xl$<H@4$|8-x 6"&K@xPh$  f~#`$?.#Ã@xx"~#?xH5$!x# xx"!cGH#;,: c0#cC x!vxX xL$!A:>xP$4x4 AYD                 @ !!! !!!   DDD  HZK EHT    UUU 222 ́́ (A4H4=`T`?@H~>H5`|dx~>Ha`|}x8ax8H!aA }"&aƨ*ʈ,̀(A 8H8̀~>8jtH`v(|&Ta~JNa~RVa~Z^a~bfaĐ~jnȐr̰vxxy|@Aqhv FϰxƇSσzO{:=d֨5^-`TqNWBX[O[$`9T KN &ˉM>e{>پK~Ht铃@'9uω.A^lO`:?S %Kz/՞gHEʧJ:}T>y51Mˍs'E^mO`:AD~©'~OiGʧJ:}T>y|514rWBCfΪӸOrikO >>yNUO^׼KTOfU`iZO?鍟RH|O^'/@ PuZftl/Sx4* ҞW`J}?%~IENDB`0`_ApSCHSN | ՜.+,D՜.+,   8'On-screen Show gI$: TimesTimes New RomanMicrosoft Office 986.3 Inverse FunctionsInverse Functions2Example: What is the inverse of y = 3x 2 ? COriginal function: f(x) = x2 Inverse: x = y2QComposition can be used to verify if two functions are inverses of each other...  Fonts UsedDesign Template Slide Titles 6> _PID_GUID'AN{8364D780-A174-11D5-9EBB-0030651936B4}`xxK8a8K`T`?AxH!A8a88K`x8!p|N |Ԑ!p|ux|#x|+x|3x bc8 8K5`|~x8a8KU`8a@H!wu^(Adx(AT`?@W>(@8` k(@8`||xxxHM`|dx8aTH`W?@4aT8D8Hp`;8aDK@!`T`?@;W?AaTH`8aT8H`8a8xH*A`H,a@xH`T`?Aa@8H`8a88@H*`T`?@xH`|dxxHQ`H`8!|N |!p8(A j(A8|+xW?AxH`|}xxxH=`LP TX\`dh $lp(t,xHdaLP aTXa\`adh $alp(t,xx(@ 8H8xW?AH `xx8LHE`~&8&8` k(@8`||xxxH`|dx8aHHm`W?@4aH888Hn5`;8a8K=u`T`?@;W?A,aHH`xHՍ`|dxxH `8aH8H`x,HE`8!|N |