ࡱ> RdO)yp Pictures* PowerPoint Document( FSummaryInformation(-6  !"#$%&'()*+,./01234589:;<=>DocumentSummaryInformation87Current User-$ Th6"&0f[E _ 6x}UKo@;}@" HpbPH= (^@BNMN*pfJݙo< KpoU:.~G19J?̑{/~%X)b\3z ߁8A^r . ju]eIj,ʢ4,me(4&IMР-. __׀,n&ֈcpMNq`N0}O麺A#h ̝ؖ J9vM4g*4 ' #Lx{?r<I1N^q/ j4`()W),z`FcdѨA8i CJuk&ƅG`s`I (i7 L> llߋXtfYgRTҨuS2O|Cg'issCEcDMкHmfmJׂAD S|K~|[0:*lh m-2sD$Ax`bf%x<{"o"s1 #&&̥\W`vÒydB REV' Jv1NyNUO^_cQ#|+iEp$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,\9qhv FϰxƇSσzO{:=d֨5^-`TqNWBX[O[$`9T KN &ˉM>e{>پK~Ht铃@  !"#$%&'()*+,(  D  0dN/0(  0;[0 0 000$([\{b00 000000000  0=] 0 0 0000 2 3 !A0C0E0G0I0c00000000000000000!%),.:;?]}acdeghijklmnopDTimesNew RomaQQ`H}^Qp}^xLNDTimes New RomanQ`H}^Qp}^xLN@  @@``   @n?" dd@  @@`` x4#    OB$6"&0fp$$b$VvE/i-!7)p `1?@=i#@ g4gdgd^}4 ppp@ <4BdBdgQQ? %^ 6.2 Function Operations&2001 by R. Villar All Rights Reserved'' Function Operations  You can perform operations, such as addition, subtraction, multiplication, and division, with functions& For example: If f(x) = 3x and g(x) = x  5 f(x) + g(x) = 3x + (x  5) = 4x  5 f(x)  g(x) = 3x  (x  5) = 2x + 5 f(x) " g(x) = 3x(x  5) = 3x2  15x What about f(x) g(x) ? Vu  == i= i=iii  >f(x) g(x) = 3x x  5 <    (Be sure to consider the domain (the possible inputs)& The domain is does not include 5 since that would make the denominator 0& Therefore, the domain is all real numbers except x = 5. Example: Find the domain of!Composition of Functions( ==If there is a 40% off sale at Nordstrom s and as an employee you receive a 10% discount, how much will you pay on a $299 jacket? You do not get 50% off& ...this is an example of a composite function. You will pay 90% of the cost (10% discount) after you pay 60% (40% discount). The two functions look like this& f(x) = 0.6x g(x) = 0.9x We can put these together in a composite function that looks like this& f(g(x))  f of g of x fii   rWork from the inside out (find g of 299 first)... f(g(299)) = f(0.9 " 299) = f(269.1) Now, find f of 269.1... = 0.6 " 269.1 = 161.46 The jacket will cost $161.468 = = =i Examples: If f(x) = x2  5 and g(x) = 3x2 + 1 find f[g(2)] and g[f(2)] S     2   = =2 =     (Df[g(2)] = f[3(2)2 + 1] f[g(2)] = f[13] f[g(2)] = (13)2  5 f[g(2)] = 164 g[f(2)] = g[(2)2  5] g[f(2)] = g[  1 ] g[f(2)] = 3( 1)2 + 1 g[f(2)] = 4( ==2==2i =2===2= i(i ~Example: If f(x) = x2 + 6 and g(x) = 3x  4 find f(g(x))F@ 2)(f[g(x)] = f(3x  4) = (3x  4)2 + 6 = 9x2  24x + 16 + 6 = 9x2  24x + 22`(' 2 ==2 = ii2 i(iP  ` ̙33` ` ff3333f` 333MMM` f` f` 3>?" dd@,|? " dd@   " @ `"  n?" dd@   @@``PR    @ ` ` p>> |( D    `4|wawa1 ?P   T Click to edit Master title style! !@  Zswawa1 ?   RClick to edit Master text styles Second level Third level Fourth level Fifth level!     SB  s *޽h ? a( &Microsoft Office 98 ZR@(  @  Zvwawa1 ? @   RClick to edit Master text styles Second level Third level Fourth level Fifth level!     Sp  01 ?    B  s *޽h ? a( P 0( @  B  s *޽h ? a(  4$( wY 4r 4 S T%` @0   r 4 S ' `     H 4 0޽h ? a( q \T0(    # l|wawa1 ?0     # l4ywawa1 ?`<$  ^ H  0޽h ? a(F e ` V(    # lwwawa1 ?P0     C %` <$  ^    <41? ,$D  DThink of the values that will make the square root a negative number ED:  <4v1? & ,$D  NThe domain is all real numbers greater than 1. The graph will confirm this...0O-i!  BA ?1?9 8 $D X   C AGrid 2 000028FD Macintosh HD ABA78158:@  a,$D :2   3 BWTCMENGpHiIWTJQo? `T3MV`T3MVWT3MVWT u ,$D H  0޽h ?  a(Լ rj (     # ltwawa1 ?     # l"wawa1 ? <$  ^ "H  0޽h ? a($ ld$(  $ $ # l4'wawa1 ?    $ # lt wawa1 ?`0 <$  ^ " $ <tu1?`D HIf f(x) = 0.6x and g(x) = 0.9x What is f(g(x)) when x = 299?"IHH $ 0޽h ? a(- rj(( @ ( ( # l,wawa1 ?Pp    ( # l!wawa1 ?pP<$  ^ "H ( 0޽h ? a(@ rj ,(  , , # luwawa1 ?P    , # lT}wawa1 ? `<$  ^ "H , 0޽h ? a(<x]OU]B"~a .P"$]!-0;Lٝ KkM4/>h 4M|<ޙeZ&ٹs={}g?>x1 DHF_ma[Bl ٷA+&΃N<7SM=3cׄImΔY; A9"Bd~,cA dsTL>as;:nxG} cb;=fyBO#> n`{pĞ>9ӈ >wr7rhh.x{}V;oh<+q'fXp*rTGb uT7لު^XX> .uڀ/bl认xacف_ p: - ^HvVnI)RqI8'q.%%uqDN6 `] Ɖdt}^.Z͠d4%KQ&J9 WmǐYFJ6&k%ub4k <+Z`B=Z,}fO봴R*mZ%d2~S7<.ԲKF['(YҞ1! Ixɨ"!ej͠Uf2huL̐:CY%GdZDJ/ʆ!y{˲b13";T8C&JBӡaK吂dFP>Oxβ!=ϐD>2BՑQ樢(N(yu-d3.cQDl3QeY7UwܵPZK`^M~ fhiʱI3trM(YUNL!a5yZւ[g2H&Fg|)_ͨ[|S^0FDE^Щu$ }# ]77CۧzIEIPY\yj3Vtj*Rڔi@՜R.HEU":Sl`!-lvfAFXu5֪Ntd,ېD \TS3ahxTF5 d^IAzn[-9qg *U(S{6u _tG/ҟ_29:77%NB/_>} Ho~":vQӱ,K|u`A~|[³P\v 30y]޿σx>.8 vr<`M#%(+P36:=&a@?E@7753W362.7819V362.784V362.7852[362.7864362.7874V362.7883T'362.7886[362.7901V362.7926W362.793T'362.7934[362.7994ͶX362.8042W362.8047T'362.8086[362.8093U362.8149Y362.8169T'362.8174[362.8176V362.8199362.8212ζX362.8221[362.8225 Oh+'0   D P \ ht|'Function OperationsuncTechnology ResourcesrPechechechSCHSolo11SMicrosoft PowerPoint 4.0oi@@@b=@@}3 GPICT P@@2Q 2HH <2 n}t:      F   L      O          H`K EPZ        9Uݽ`ʆ ٻNKX\mHš(_ '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`D 4D |>]@_EmSCHSl2б8vfH,mI%&S%Pťw +;\+$R^x>Klv3_4P{^}컩,7!@mz  } ՜.+,D՜.+,(   ^'On-screen Showf7Q4:  TimesTimes New RomanMicrosoft Office 986.2 Function OperationsFunction Operations f(x) g(x) = 3x x 5 Composition of FunctionsNo Slide TitleSExamples: If f(x) = x2 5 and g(x) = 3x2 + 1 find f[g(2)] and g[f(2)] @Example: If f(x) = x2 + 6 and g(x) = 3x 4 find f(g(x))  Fonts UsedDesign TemplateEmbedded OLE Servers Slide Titles 6> _PID_GUID'AN{D93F4281-9F1C-11D5-9EBB-0030651936B4}CץQ|5ۖk /U]&4jF>txmk𣜤 AهS oq f8CX֠ORO'ZN:K,9` 2X D {AG[ 9'iNLmD0E!e&f F%0yA&SAfΤI6iXǗ-kG kyS_=[$BE&X)4& )_cؕDPY `fVrdYchr eŇh4*p&WZeu<,*NNAlmv IےS$)&4VOār\.vR u`"UӂrRUa\ ePJk/N6ΧRLu ԔD+&nN  t8qՀTG U$S2vC_8zh#&#[&GXZx e~N!ڎ*pνЙ(Ʋ8%QLV@u2wв \ٴ=KgK&Fu!;{y '/,&$QIZ hr'l+kҵ6%&Eekfgvj#˸"pr⣑d0S3 IP&Y1A # /ڠ!C ˻ck~\ G\>Enڗl],$UI"QH'vA®[X PmήK+WUkX""HeHɖ%, M|%m..S7n#d[<٨a1ePz@&kS,&XI0 NBbQn:ڎu馮W+S%wj5sH BCo ' X'5)&W,DO,WWqJuS҅PBL #U^s# UƧ #Ki ،r\.( -dp3ہĝT2\8XEU Lg,u0&R.B>ؙ@)dծO*틄+yL,]9e&m/[ş辚򠕦)_DutM4Oѽ&8&Cಖ4]rOШZmvd"U.BfQT)T4ex{d Կn0`5qFҭ 2HQFͿNKmǿK% NްLېX bY 8 &Z#76%׈d:@D ` k zf>y)jGBƶ&rB F-V~""57 s̿Meuk#e4c&#ߢ! 9 pښXDczTs:Gx$Lغ^Xq$0.{Y_:n7vd*f*%-)E|f+O鸐y]NȀG#X=Uu,2m;b9S ݼ:ЁC)4 ~aK0`턃(@Z/" NFHXv< A<긑wa0y^D~g e-h[|