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# Obj. 14a-16b MAC1105

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This 11 page Class Notes was uploaded by Michaela Maynard on Saturday October 17, 2015. The Class Notes belongs to MAC1105 at Florida State University taught by Pennington LeNoir in Summer 2015. Since its upload, it has received 43 views. For similar materials see College Algebra in Calculus and Pre Calculus at Florida State University.

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Date Created: 10/17/15

RV another Obj 14a example If 30 is a point on the graph of y f 13 then which of the 39 7 followmg must be on the graph of y f ffja cal X035 3 a 66 03 0 3 3 0 WM another Obj 14a example If 30 is a point on the graph of y f 23 then which of the follo in must be on the ra h of ac w g g p y f aged across zaams 03 0 3gt 30 lk another Obj 14a example If a function f has Domain 2 0 then what must be the Domain Ofyz fW eflcm across lo39asm q sleDomam 02 23 does not 39 Change another Obj 14a example If a function f has Domain 20 then what must be the Domain ofyfcVRHgtec am a ams 201 23 another Obj 14a example If a function f has Range 20 then what must be the Range of y 13 Ken at 004955 Ioasas v A f Do 63 offf Lt Q Q 2 01 27 21 a n5 e another Obj 14a example If a function f has Range 20 then what must be the Range of y Cmy K leu 0WD5 giants I Doc nor 0 2 42 cm v13 e 0H3 t Obj 14c Graphing With Vertical and Horizontal Translations How do these compare yf93 yf3700gt0 yfc ccgt0 We will consider a speci c example to justify the general case mam 43W 2 5 an xL xiii 0390 75 3 5 I l H q 5 How do these compare yfm yfxccgt0 We will consider a speci c example to justify the general case yzx26 ym y sic 3 Smx39b y y gamp 3 Qm0 939 493150 R 01 Obj 14c example Select the function whose graph is the graph of y 35 but is shifted left 3 units down h up 92535 3 ym35 yzx53 63 0U 631 Pl60wh mead V5OQ H t name Ob J 14c example For y f as de ned below graph 3 f as 2 and Slugb V 5hrl39k 35109 2 3 Obj 14b Graphing with Vertical or Horizontal Compression or Stretching How do these compare yf1v ycfmcgt1 ycfx0ltclt1 We will consider a specific example to justify the general case 17211321 R39 013 d39 CHUCK yJ7 Ll L37 H44 10 MR U z rm O PCWUQ How do these compare 921 yf0cgt1 yfcx0ltclt1 We will consider a speci c example to justify the general case 212W PM were RevaW S aV 0 M2 RWMIIAS 3 kW 30 Same 495342wyxae n39value 3 neeoco co mahe 9 so 3 v Gums For 3 f m as de ned below graph 3 3 fx and y f M M W LIW39ECfoPtS A3 VCY b howl E I I M I I 1 x T VVWCC W5 C WNW 3quot Wd Rahgci 0393 4 3 Domam 47 33 OI 1 tRemams tIu mam w I Jame Ea ML RA another Obj 14b example If 2 4 is a point on the graph of y f as then which of the following must be on the graph ofrowe j 44 4 8 lt1 2 lt2 4 28 O 56 value dime a 9 H another Obj 14b example If 40 is a poing on the graph of y f x then which of the following must be on the graph ofw T an ff 8 0 2 0 0 4 08 0 2 W Mult g valwc by 65 another Obj 14b example If a function f has Domain 08 then what must be the Domain of vanrm r y4f quotT39ancr aHeccs 032 02 Grange b9 Ucor o q 1 Mult y value by Ll another Obj 14b example If a function f has Domain 08 then what must be the Domain of y f 456 P FCLBS Narmwcr a 032 08 t W 9 Pach G Ll MMH2 Lrvalufs log 3914 another Obj 14b example If a function f has Ran e 08 then what must be the Range of W Tana OHCL Q quot I 02 08 tw angle f 1 J Mqu q vmws log 4 another Obj 14b example If a function f has Ran e 08 then what must be the Range of yf4 Nawowcr affect3 032 02 km domain KaiMy 6063 Change RR Obj 14 Summary Obj 14a Obj 14b Obj 14b Obj 14c Obj 14c Affects Range y myquot across KrauS y3 m 0quot taller yi w 0464 Showon 3f v2 ohm H yfrc5 SW4 d wn TEEQEEELQX yfw Rm across 301 yf3x 07 warmer yfiv 04C 4 wMEr yfrv2 Sh 39t H45 yfm5 Shr t jv t to51quot1539 Obj 15 Operations on Functions forming new functions by adding subtracting multiplying or dividing existing functions De nition of notation used to denote these functions de ned for all 1 in the domain of 7quot f gm 2 30 3 0 f 9W FCXD 3 0quotquot r fgx PM o 5063 x Z 3 8 Obj 15a examples frr 21 yo N ha 3 mltxgt x Find 9 716 Intege or exact de 39mal only mathematical operators are not allowed mlu Clio mLo 9 U 3 b 20 Z 3 an quotLo 392 3xo Find m h0 Integer or exact decimal only mathematical operators are not allowed Nx0gt hm o 3 3 RR Find 4 If noninteger answer give fraction or exact decimal Don t give decimal approxi h mation 2 4211 11 M43 9 am quot 3 U 3 393 quot 3 Find h f3 If noninteger answer give fraction or exact decimal Don t give decimal approxi mation a a h 3 O 2 39 O a 0 193 3 33 35 3 Obj 15b examples Find the domain of a sum difference product or quotient function Recall that these functions are de ned for all an in the domain of i and 1396 OVCFG t1 0 Y dOCS n 0 ma tth For f3 V211 8 and 93 2 39c 2 nd the domain of f g 320 c2gtCgt 07603 cgt2 D 0 P9 3 l Ll 23 6nttrsech0h 4 f 1 N For f 3 1 x and gc 1 41 nd the domain of f g Ix O Lamas D o 3 696371 I Z 6 l 2 LIX quot L39 2x W 1 la For fxa 4and gxa nd the dom ln offg M HiO Xv20 D 01 Na 3 unde ned z 3L gtlt i 4ND Liz393 4 TM ND vxte ectwj 51A 1179 vmu For fx 2 33 9 and ga Eai S nd the domain of ftlg 5 gt 3xqzo gs 0 D 01 99 5 02 5 3KZ39q 3 AW 5 WV 395 r 3 quot For fc 32 9 and gx nd the domain of f g ag920 airs20 TD 03 j OolBJ 3K3q am 55 Xi39vj gND K5572 W 3 E A 393 53 For fx33c 9and gx x 5 nd the domain of f g 39 Cc 5 D 03 Ric60 D 0 if Eki fag an5 I W 1 1 a 573 For 1653 and 956 nd the domain of fg 973 9 2 5 odd 3Liq f0 025O Domain o t 3 00 32 3Kq 0 655 U23U 3 0gt 3x 5 L55 0R 3 my 7 W MMEMSB T i 3 u Obj 16 Function Composition Given two functions f and 9 then the composite function denoted as Q oq is de ned as follows foga 63gt for allxin the domain ofg such that 91 is in the domain of f cw Q 33 5 than 5 hOS so 06 m cm domain of 3 Obj 16aexamples 3 D h Jcm domawx fx1x 93x2 2 o 9 Find fog 2 Find9 f 2 It mam 309m 9ltT9 5 399 5 3 3355303927 339Jz 8392 quot3 7 M Q Mw39b 3 Iqgt quot73 IB aa TL 6 CWOKLC 39iU Iquot Find f 0 9M N900 Peta a l quotquot PM I d Max3955 Find fof 0600 4amp3 I l re far o 39quotquotquot l x Findltgoggtltx 33003 33 2 739 33 19 2 335L2392 gt4 351 seq 263439Ll 4agt 07313 30 27x 3on3 tax pla Lisa 1 27y 31 136 6 AfMMlb CW9 C Obj 16b Find the domain of a composite function for two rational functions Plan of Attack For 9 O 5 g Method 1 This is the simplest calculation but requires an understanding of the de nition of function composition kigt0 3203 P than g 3 twat nc awhf d sheh gt06 5 rst Puncdpphed a Determine the domain of the rst function applied b Determine if the second function has any restriction any bad value c The rst function a the bad value of the second function So set the rst function bad value of second solve for any additional restriction Method 2 It is easier to remember this process but requires simpli cation of a complex fraction a Determine the domain of the rst function applied b Form the composite function and see if any additional restriction KO For fm i and gx 1 zit 2 Find the domain of fog Find the domain of g o f D M oi raj 2x mag 9 01 30 inlm omggg hr 0 aw 3gtx7 o 3 quot31 b bad value bra 3 a b bad value 0 t S O cvrt gt71 C330 gt0 cho 5 9 I5quot I OxxR x glza l 0 L 2 ya dchbowa ND eesbncbiov 30 x 74 ill For fx 2335 and 923 3366 Find the domain of f o 9 Find the domain of o f D O 33 ELILfRL I03 9 0 r 30 146755 a For 9a mm a ear 1L0 5 5 b bad value 032 4 is 5 17 bad vawc ol 3 b 8 C 9605139 CD C0 R if 7 pg 6 5 gt 06 b ArcA i A gt 07quot 739 3K 5 Rx RxIb 06 EIL I ixa Io O I39O 3 quot 4am Odmhonal rf sicmach This next example illustrates that the domain of the composite function can never be any larger than the domain of the rst function applied Forfxx2andgx 4 D 0 03 X 2443 Find the domain of fog UK SOCgtIK Z L bad value tor I3 home Smce dowwm must be Composed 04 we Pump 1

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