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# Exam 2 Study Guide MAC1105

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This 29 page Study Guide was uploaded by Michaela Maynard on Saturday October 17, 2015. The Study Guide belongs to MAC1105 at Florida State University taught by Pennington LeNoir in Summer 2015. Since its upload, it has received 233 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

lth S Chd 3th gXQm a Obj 100 Domain and Range lee Domam and Range from the graplil of a functlon D EHJ39J DIM new ij Domamar valu632oo4 c v H m 00 DOWNMAW A anngl vaws 5 34 m Pwnc haMOf j K R 02f 3U C 3 0 R 00gt Exam Two toll 5 More Obj 10c Give Domain only from the function rule We re being asked for the la set of real numbers that can be substituted in for g What prevents a value from being plugged in for 3 When might the domain not be all real numbers 531 There are only TWO situations gat limit restrict the values that can be used for x On 0 w h w V w 3 u c WQUVC For L W0 k quotdcr radical 65 46M Q unCtWY rulf dOCSVH vxvov t 05 thine than 0104290 Give the Domain f9 3x7 5134 395 923 3 52 17 ha 31quot 960005 Dzewlw a Di39 01 0 fx F3quot 9 have be poi 36 7 2 0 3x3 Some on line problems ask For f V33 7 select the values that are in the domain of this function I Wm M valu 63 SC 6 9 awg P Choices 5 6Q 39LUOY quot IAqu POS 905 W a quot dimrfi M 3 CaCuatc lOMOHn choo3e 3K730 322 7 valuf S in dode C 2 quot 73 00 Give the Domain fx c g 75 a7 34 0 3k 239 R Drr39w39 31 kquot 93 anan the vmw fb zor 5 that wound caut c du gOn 7 Give the Domain m E2 10839 039 50 x43 5 n93 g 7 domain 39 O V K 3L 3 TM 3 61 09 a K s ekce t K D 03gtU339 0gt 0R EblLfZE P 5 2113 3 Give the Domain fx cf39b v 3quot 39 quotRWa a p b37 i gt V I c 22 OR Shoo 1c LN M domm D EL x 2 22 Give the Domain fx 258 3 lagM 0 9 OOJQOB 841 No x1 Give the Domain fx 2 7 m quot LQ L Lo 0 2 D ogug3u3oo QC 4 za 0 0R Lgo K1350 31quot 1 Edna 2 U33 A K 4433 Give the Domain fx 2 7 x2x 1 K2A 50 Diik x i E k 006 0 0 D gimi or 6 4 Jig MOOQ Y 52 x UV a 763 36 Some online problems just ask Is the Domain all reals or not all reals When the denominator is Quadratic you can use the Wfrom the quadratic formula W to answer wean bL 1 C 0 31 Solvci 39 b1quot VZbE la I cc 9 2 K C Domain all reals f 3 x2 x 1 f n Ra baa Hac l i 3 ba Mac 1 Q l 394 mob a reals quotNo c quotan ream cc1 m2cc1 badlac lad H00 2 lfq 7 340 quotPm rcols Domain all reals f x 7 Domain all reals my 39539 m 5 W to 39HOC K 2440 7 gquot 3JcquotJlt5 39Not a reals k c a b c 43463903 pr 20 39Not all rtals Domain all reals fac Eff 3T6 fa 2 4 logMac 7 quotla9U 9 WI 9 t l aq 23 40 quot0H fecllS39 Domain all reals fa p xdkl x 2 a1 7 2 3JX2 3 go 039 xxal 0 0 0 6 2 gt 0 1C a V 0 LBC V C Ch quotNot a mob x 752 Dquot Elixiag 37 9quot9 1w Obj 10d Find information from and about the graph of a function There are several different types of problems in this Objective 161 Obj 10d example If fx 2 Q ZC CZ l iSthe point 31 or the graph of y fa 3 l YES or Is QC f a abA 32339 z 5 7 GM 0 Pomt W N W 3L 0h tm raPtx39 H M33 1quot 5 3 30 Obj 10d example For f 2 2 gig if f 51 2 what is x ml 29 6quotc 5 K 29 b 3 y Wl39xKTvSl 35 K 3 K 0 none of these 7 Biff quot7 a 7 x Obj 10d examples h For the function shown g 7 D graF For the function shown 3 4 O 8Y0 h for what 2 is f gt 0 W for what a is f lt 0 l3 above t 33 l5 below 10 Cum Lalm 3 233 Ll LD 9m 0609 x x U 00 Obj 10d examples For the function shown For the function shown for what 13 is fv gt0 3 7O 5 abovC for what a is fxlt0 3 A O a Y 3 m 5 5 b a m below L Cll lb x PMquot 3 3 HSn 2 3 a U 38 xO Obj 10d examples whev 3 3b7 S quot For the function shown 3 7 For the function shown Y a c if m 0 what is fv what N o for what x is fa O 6 eta394 130 3 U Yt PampJL Obj 10d example Answer the questions below for the function shown V f 1 is positive negative when bc l a 3 39VDOES not QHSt 0 55 f 2 is positive negative 64 15 tm asc mpw ce rhhe mevtv twang f 25 i negative unde ned f 3 Q f4 L How many times does the line 3 5 intersect the graph Mg Copyright 2010 present Annette Blackwelder all rights are reserved Reproduction or distri bution of these notes is strictly forbidden 39 Obj 11 Properties of Functions Obj 11a Increasing Decreasing Constant Always look if lll to ght as x values are getting larger The function is ac if the y values are getting larger The function is 66 C if the y values are getting smaller The function is C OY 3t if the g values don t change lllllll 35 45S 00 3 65 Increasing on 3 7 Increasing on w Decreasing on quotgt0 i 3 V 7 00 Decreasing on 39w Constant on Q Constant on kl Obj 11b Local Max Local Min A local man is a point Where the function changes from VC to 6 EC A local WY is a point Where the function changes from 60 to MAC Mt 5 a The maxmin IS the 9 Maude w W w l mun The maxmin IS AT the J Value 0va 5 7 MALm1quot An L V y R78 P2 4 xv Q24 For point P The local Maw For point Q The local Wmn For point R The local gt X is Ll is quot Ll MM is 55 what 5 We qrvalu 397 6 wmrc 3 t39 5 Wm The local mm is at a The local mum is at x j The local W l6 is at x l Obj 11c Average Rate of Change O 9 5 The average rate of change 011 between c and a is the of the secant line between c fE and 51 fz r 3 Cant me Comn66t5 a Average Rate of Change 2 1 two 370mb 0 L C a h MCMofVLC a 9 Formum Ty WRL DW The average rate of change the slope of the graph at x c In calculus the derivative gives the slope of the graph at a c which is the slope of the tangent line at a z c Plan of Attack Set up the two ordered pairs 0 f and x f ll in the values then just nd the slope Always REDUCES A1 Obj 110 Example Find the average rate of change of f between 3 and a for f as 3a2 2x 5 o 1803 3 3333 3 5 R39I W5 33 5 33 W 9 239va Rate Bl Gwaan 003ch 0 0 3239 2 5j 6 C 30quot in 5quot 3933 3x 2a f j 5 an 33 quot x3 X 3 x3 2 33 Bic 0643 31o e 139 7 v 3 2 Obj llc Example Find the average rate of change of f between 4 and 1 for 04 if t 433 HI aggt 5 quot39 quot quot39 202 23 Z39 2 4 ivy Rafe M Chan b 1 T A p K gt 3 b C L quota 2K quotquot2 L K 39l l 3042 f 39 d 2W zq 3n39l M 39 Ubj 11c Example Find the average rate of change of f between 1 and a for f I2 2x R r 2 1 03 26 M 5 54 Jc 2x a a 3 a 3 L32K3 Hv Katt 0Q O M V T 66Q Qquot 47 K 3 F f K H L C KZ R U Q Viv 3 nl m quot Obj 11c Example Find the average rate of chan e of f between 4 and a for f 13 21 mpg Hra we2 no 80 09 Oman 39 H0462 94 21 a 1 at a all 104 1 0 339EI N aaw M l L Ll m w 464 v i quot aw JV 27 L gt Mule A C howe thCC39irmc rationalize numeracowr k WW Obj llc Example Find the average rate of change of f between 5 and 2 for fa 2 2J5 The square root function examples will quiz only not test 5 86 am to W M c Me It to Avg Ratc olquot Chanj 3 4Q 11 a qgg an NS 3 qu 30 40425 L C 71 quot5 EVEng I 6 53 3 539 r Do Noe mutt ow MGM 25 q 2 ML jarfrail 1 Z 17445 Checc Obj 11c Example Find the average rate of change of f etween 3 and 1 for f at xx 1 The square root function examples will quiz only not test 93 WW a 35 mm Irma Rate 0 Cmngu 1900190 27 2 WET r22 K C 39 L g mi9 7 10quot 39Ll M Do 40 malt ouel ZD f 3gt W 39 W Mult Chance Obj 11d Slope of the Secant Line This is the basis of the de nition of the derivative in calculus The Ob 11d problems gill quiz only not test 5 9 LA 7quot Find the slope of the secant line between 1 f x5 and a h f a h for fc x2 21 3 3 m 3 r w h My mm 4 n quot NW0 MD 3 W3 3 x h so h 242 3 KRmhhaah fK w m 26 1 n h snowman s MC3 Rgt a h9L crfs h K x No Cayman Hamilquot Find the slope of the secant line between 13 fx and a h f w h for fa a2 3 New 39uvb za x 11 73 km a 3xh39na393 3 39 war so V 3 a 3 WC jggzmm 33 2 LaRlbh Ma 3 m x I A uhx RW V BW h dbl 7 T T mr h thamp irhh 93 Obj 119 Symmetry Properties for Functions 3 5 3 Qio Er 1 y M 0 VODMCt I n 1 KI L x IL x39 w ODD No rail 15 Graph is symmetric Graph is symmetric 38 mm c Pbr with respect to l39 0 5 with respect to Q 5l Qu nchov flt agt 19L fxgt LEE Plan of Attack To test for Symmetry nd f 50 If f x then function is m If f 513 f then function is Q If f v 2W then function is Mix CVCn of Odd Obj 1 le Even Odd Neither fxx4 3cc25 fxx7 75m53a 5 39 1560 5 Oq3C z7 r5 Pquot quotquot7 73971quot 36quot 39 0quot 3 6qu2 5 2quot 5x5quot3 75H ev n s M 7 Invcoww mhan Qwe 39OM fZ l L74 5b53p gt quotFCQ fxx5 5x2 fc 2x22i3 3 cm 405 5916 w am 31 Ma a t wag 3 026074 2V3 not CVC n 3 we 1w w L rcaraf Mt 1 00 nob We wooing odd 39 9 quot0396 Udd 55k fx35 fa14 3 L nzi 1 t amp zx not 73 we 3 X quot x 21quot amp2q quot even o hO39t QVCY AA a I 39 Nanak ed ltxgt xq NW quot F Z gt 23 Tiquot 9quot 04 odd Obj 12 Collection of 7 basic functions Plan of Attack You DO NOT need to memorize the properties Learn the GRAPH and the FUNCTION rule You should be able to answer any question if you know the GRAPH 1 Special case of a linear function f as 10 y 9160400 2 30 4 0 odd 9on055mm wrt Ofl lh thYEOShij on 39wlw 2 Squaring function f 513 932 wquot D OOIODgt 0 w aCvx 53mm wrt 9 4quot 5 DECICOSn3 IMHO Manama 01 00 3 Cubing function f x3 9 4 91 00 R of Odo ngm Wt 006er intvCaSnrx on 3900 0 AK RV gt39Hr Z quotl o l 4 o x t4 3 395 7 3 0 o J L a 7 M eve v63 4 Square Root function f cc NO 38mm0 am 3 nexmev odd nor even N on 000 020 065 f 5 Cube Root function f x Nab 175 wew odd lune R 4 3 mm m 09V nC On oo 0gt 6 Absolute value function f as a l 39039 quot 5 RI 0 co Even PLAN 3de W f o y aw w ok DEC 0 MIO C on 7 Reciprocal function f 4 K 0 91zw0uo 0gt a wow wow 7 tuft ng decr 000 MO b AR Obj 12 example Enter the equation of the function whose graph is shown Enter both sides of the equation not just the right hand side You must use 3 on the left side of the equation f a is not owed in this 103639 V sMust MCMOI IZC Fume quot Mora for thwi arr 2 W WCs on came Your Answer c 9 l l Dcm It 90 EaventhCMS Ob j 12 example Select all of the following that are symmetric with respect to the yaxis There may be more than one 311 a 3PM x3 PM 3 fx 35 ms 5 lt35 Obj 12 example 7 5 Select all of the following that are true for the function f x ac There may be more than one ltI Domain is 0 00 Domain is 00 O U 000 Range is 00 00 Range is 00 0 U 0 oo raph is symmetric with respect to theyg Graph is symmetric with respect to the origin 763 Obj 13 Functions de ned piecewise A function de ned piecewise has a different function rule for different parts of the domain Obj 13a Evaluate a functions de ned piecewise Plan of Attack Compare the xvalue with the restrictions on the right that tells you which function rule to use DO NOT plug as into each A7 x2 ifxlt3 fc 112 if3ltx5 O ifasgt5 Find each of the following f1 l r R f301 095 90mm f5001 O 1 if x S 1 fx 0 if 1ltalt1 1 if a 21 Find each of the following f 1 x a q H933 1 0 f3 3 5 f35 3 5 R 15 mm O f05 o Obj 13b Graph a functions de ned piecewise A warmup question Graphfxc 2ifx2 1 A g quot74 x T a 1 ng Value hot 0 Pm Pfr AunChon f5 53 Graph faa Qifagt1 II 4y K 2 AR quot l 0 01 Rf Graph f52x ifxlt3 K 7 O 5 1 x43 th 95 Ax 3x 1 ifxlt 1 Graphfx 4 ifxgt 1 Ay Ma a r X H CWCIQ ovt OPB 6 U 4 own male 4391 3 b Ix 0 39 lLl x4quot twn 6 i man 8 3X 5 3 4 4 39 ifxgt0 3 Graph m it i x u i 3Q twat ovc a 2 4 f lt0 v60 oven cwue x X 0 LI 0 R H x 20 I k 40 lme MWEVCW GOUNd I 9 5 th j 4 a 44quot behawm Hum 3 5 AG 333 5 ifxlt2 Graphfm 2 ifxgt2 2 R 393 b H 399 l gt643 M Li 46va 33Lv5 U g Obj 14 Graphing with Re ections Compressions Stretching Translations Obj 14a Graphing with Re ections How do these compare y fx y fx 3 fv We will consider a speci c example to justify the general case y5 czo yE 530 y x LZO 1c f O y across y Krams Aycztxoss i 030 grams g R1 0001 1 j x I x 5 D39 00 O y y R 0 w 011000 X IZ fl 6 o 0 Rquot 0 w 0 4 x l H 3 2 O O quotI l q a m 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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