Math Review for Algebra and Precalculus

Transcription

1 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Mt Review for Alger nd Prelulus Stnley Oken Deprtment of Mtemtis Te City College of CUNY Copyrigt Jnury 00

2 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Tle of Contents Prt I: Alger Notes for Mt 9 Introdution.... Bsi lger lws; order of opertions.... How lger works... Simplifying polynomil epressions 9. Funtions.... Wen to use prenteses.... Working wit frtions. Adding frtions...

3 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Mt Review for Prelulus nd Clulus Prt I: Alger Introdution Alger is te lnguge of lulus, nd lulus is needed for siene nd engineering. Wen you ttk rel-world prolem, you wnt to represent te prolem using lger epressions. Wen you red tenil ooks, you wnt to e omfortle deipering nd working wit tese epressions. Computers n t do eiter of tese tsks for you. Alger used in undergrdute mtemtis involves tree min tivities: rewriting epressions, solving equtions, nd solving inequlities. You need to perform tese somewt menil tivities quikly nd urtely. It s very diffiult to ieve tis gol unless you understnd ow lger works. Alger is symoli lnguge tt llows ommunition etween people wo don t know e oters spoken lnguge. Te grmmr of te lnguge involves tree min omponents: epressions, identities, nd equtions. An epression involves numers, vriles, prenteses, nd lger opertions. Bsi types of epressions re integers, vriles, monomils, polynomils, nd so fort. We ll del mostly wit epressions in one vrile, su s te polynomil. An identity etween two epressions, written wit n equls sign, is sttement tt e epression n e otined y rewriting te oter. A simple emple is. Wit rre eeptions, sustituting numers for vriles turns n identity into true sttement out numers. For emple, setting to yields. An importnt prt of lger is using identities to rewrite epressions. An eqution is lso sttement tt two epressions re equl. In most equtions, owever, equlity olds only for speifi vlues of te vrile. For emple, te sttement is true only wen is or. We sy tt te solutions of te eqution re nd. Plese rememer;: we rewrite epressions ut we solve equtions. In tis preliminry edition, setion edings su s AN re used for Alger Notes, Setion.

4 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. CHAPTER : BASIC ALGEBRA LAWS; ORDER OF OPERATIONS.. Alger opertions nd nottion Let s egin wit two triky emples. Emple..: Rewrite s n epression wit no negtive powers. Rigt: Wrong: Emple..: Simplify te epression Rigt: Wrong: To understnd wt s going on, we need to review in some detil te five lger opertions: ddition, sutrtion, multiplition, division, nd eponentition. E of tese is lled inry opertion euse it is used to omine two epressions. Te tle elow lists nottion nd terminology for tese opertions. Te lst entry sows speil opertion lled negtion, wi opertes on one epression nd is n revition for multiplition y. Opertion Write Sy Desrie te nswer Addition plus Te sum of nd Sutrtion sutrt or minus Te differene of nd Multiplition Trditionl nottion times Te produt of nd Trditionl nottion * Clultor nottion Implied times sign y Implied times sign Division Seldom used / Clultor nottion Trditionl nottion Eponentition Trditionl nottion ^ Clultor nottion Negtion divided y four tirds over slng rised to te rd power Negtive or Minus Negtive of minus Te quotient of y Te rd power of is te se. is te eponent. Te negtion dditive opposite of is. It s it nnoying tt te minus sign is used for tree different purposes: nming negtive numer, sutrtion, nd negtion. Speifilly:

5 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. A minus sign efore numer s in is prt of te numer nme. A minus sign etween two epressions s in y mens sutrt. A minus sign tt is te leftmost symol in n epression s in, or one tt follows left prentesis s in, mens negtion: multiply y. To redue te numer of menings from tree to two, some ooks reserve te long ds for negtion or sutrtion ut write negtive numers wit sort ds e.g., s -. Te fewer symols n epression ontins, te esier it is to understnd. Te multiplition sign is left out s often s possile: etween digit nd prentesis, nd etween two prenteses, s sown in te emples elow. Multiplition sign omitted Multiplition sign inluded. Evluting Numeril Epressions Alger egins wit ritmeti nd te properties of wole numers: 0,,,,,,, Te most si property is te identity lw, wi sttes tt dding zero to ny numer or multiplying it y leve te numer unnged. One importnt property of ddition of wole numers is tt ddition is ommuttive: two numers n e dded in eiter order. Some emples of tis property: To tully stte te lw, we use letters nd to stnd for ny two wole numers. Commuttive lw for ddition:. Anoter importnt property of ddition of wole numers is tt ddition is ssoitive: wen you dd tree numers, te two possile metods of grouping yield te sme nswer. Some emples: To stte te ssoitive lw, we let,, nd stnd for ny tree wole numers. Assoitive lw for ddition: Te ssoitive lw llows us to write epressions su s nd 89 witout using prenteses.

6 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Togeter, te ssoitive nd ommuttive lws for ddition llow us to write sums witout prenteses nd to reorder te numers in te sum. Tus et. For emple, using ; ;, we ve. Similr lws pply to multiplition: Commuttive lw for multiplition: Assoitive lw for multiplition: Togeter, te ssoitive nd ommuttive lws for multiplition llow us to write produts witout prenteses nd to reorder te numers in te produt. Tus et. For emple,. Addition nd multiplition re onneted y te two versions of te distriutive lw, wi sttes tt wen sum is multiplied y numer, te multiplier n e distriuted mong te numers tt mke up te sum. Distriutive lw: nd Te si rules, or lws, of ritmeti nd lger re listed in te following tle, in wi,, represent ny wole numers, ddition is indited y, nd multiplition is indited y. Bsi lger lws for wole numers Nme of Lw Sttement of lw Emple wit ; ; ; Cek te emple y working out ot sides Identity lw for Commuttive lw for 0 0 Assoitive lw for? 9 9 Identity lw for Commuttive lw for Assoitive lw for? Distriutive lw? 8 In te ove disussion, we sid we were working wit wole numers. As you lredy know, tere re more generl lsses of numers. Integers signed wole numers re,,,, 0,,,,

7 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Rtionl numers re frtions of integers:,,, Deiml numers re rtionl numers wit denomintors, 0, 00, 000, : Emples re., 0.0, nd Rel numers inlude rtionl numers nd oter numers su s e, π, tt nnot e epressed s frtions of integers. Oter lws nd definitions tt pply to te tegories of numers listed ove re: 0 : Te sum of numer nd its negtive is zero. Sutrtion is defined y. were,,, d re integers. Here nd d re not zero. d d is lled te reiprol of if 0. Te produt of non-zero rtionl numer nd its reiprol is. d Division is defined y if,, d re not zero. d To divide y non-zero frtion, multiply y its reiprol. Eerise..: Using te tle ove s model, plese fill in te following tles. Bsi lger lws for integers Nme of Lw Sttement of lw Emple wit ; ; ; Cek te emple y working out ot sides Identity lw for Commuttive lw for 0 Assoitive lw for Identity lw for Commuttive lw for Assoitive lw for Distriutive lw

8 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Bsi lger lws for rtionl numers Nme of Lw Sttement of lw Emple wit ; ; Identity lw for 0 Cek te emple y working out ot sides Commuttive lw for Assoitive lw for Identity lw for Commuttive lw for Assoitive lw for Distriutive lw Bsi lger lws for deiml numers Nme of Lw Sttement of lw Emple wit.;.;. Cek te emple y working out ot sides Identity lw for Commuttive lw for 0 Assoitive lw for Identity lw for Commuttive lw for Assoitive lw for Distriutive lw 8

9 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor.. PEMDAS nd Order of Opertions We just sw tt n e written witout prenteses, euse te ssoitive lw tells us tt. Similrly, tere re mny wys to insert prenteses in, ut te ssoitive lw gurntees tt te nswer will e te sme in tis se, 0 no mtter were te prenteses re inserted. Very often, owever, te oie of prenteses mtters. For emple, 0 ut. Wen you onstrut longer epressions, te sitution gets worse. For emple, wt does 8 men? 8 Some possiilities: [ 8 ] [ ] 8 [8 ] [8 ] 0 To resolve te miguity in su epressions, mt people ve speified order of opertions rules tt tell you wt opertion to do first in n epression tt doesn t ve prenteses. Anoter wy to sy tis is tt order of opertions rules sow were to insert prenteses in ritmeti epressions. You proly know te ronym PEMDAS Prenteses, Eponents, Multiplition nd Division, Addition nd Sutrtion for order of opertions in numeril or more generl epressions. PEMDAS is usully desried s step-y-step proedure for removing prenteses from numeril epression. However, it pplies to epressions wit vriles s well. One wy to eplin PEMDAS is s follows te word evlute mens find te vlue of : To evlute numeril epression tt doesn t involve prenteses: Step : Evlute ll Eponents powers, working from rigt to left. Note: In te following disussion, we will refer to eponents s powers; do not onfuse te word powers wit te P in PEMDAS, wi stnds for Prenteses. Step : Evlute ll Multiplitions nd Divisions, working from left to rigt. Step : Evlute ll Additions nd Sutrtions, working from left to rigt. To evlute numeril epression tt does involve prenteses: Step I: Find n innermost set of Prenteses: pir of prenteses tt don t ve oter prenteses etween tem. Evlute te epression etween tem using Steps,, ove. Step II: If ny prenteses remin, go to Step I. Oterwise you re done. 9

10 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. You don t need to egin wit n innermost set of prenteses: you n simplify prentesized epressions in ny order you like. For emple, to simplify te epression, you ould eiter egin wit te inner set of prenteses: or egin wit te outer set of prenteses, otining te sme nswer: 8. Furtermore, you n work on severl sets of prenteses simultneously. Tis is done in te rigt olumn of te following emple: te left olumn works on one set t time nd follows te PEMDAS rules stritly. 8 8/ / 0 8/ 8/ 8/ / / / / / / / Reding epressions wit minus signs: It s wort mentioning few triky emples involving minus signs. Te word minus nd te word negtive men pretty mu te sme ting, ltoug te prse minus numer is not used. Don t use minus to men sutrtion. Symols Words sutrt equls sutrt equls minus 0 Minus sutrt equls minus 0 Minus sutrt minus equls times te quntity: sutrt : equls times sutrt equls Here re some simple emples wit numeril epressions. In e emple, we egin y inserting prenteses round te opertion tt omes first s speified y te order of opertions rules. 0

11 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Powers efore ddition 0 Multiplition efore ddition 8 Powers efore multiplition 8 Do powers from rigt to left Add/sutrt from left to rigt 8 Multiply/divide from left to rigt. Te following emples omit prenteses implied y order of opertions. 0 Add/sutrt from left to rigt 8 8 Multiply/divide from left to rigt 9 / Multiply/divide from left to rigt 0 8 Do powers from rigt to left top to ottom 9. Multiply efore you dd or sutrt Multiply efore you dd or sutrt 9 Do powers efore nyting else. 9 Do powers efore nyting else. 9 Do powers efore nyting else. / / Do powers efore nyting else. Eerise..: Use PEMDAS to simplify e of te following numeril epressions. Strt y inserting prenteses tt re implied y te order of opertions. m 0 8 n 0 o d p 0 e /8 q 0 f r 0 g 0 8 s 0 / / t i u j / v k w 8 9 l

12 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. CHAPTER : HOW ALGEBRA WORKS.. Implied multiplition signs in epressions involving vriles Te multiplition sign is left out s often s possile wenever it ppers etween two vriles, etween two prenteses, or etween ny two of te symols: digit, vrile, prentesis. One eeption: te epression, wi migt resonly e tken to men times, is not used; write or, preferly,. Epression wit multiplition signs omitted Signs inluded digit followed y vrile y vrile followed y vrile y vrile followed y left prentesis vrile following rigt prentesis y digit followed y left prentesis y y digit following left prentesis y left prentesis following rigt prentesis y eponent followed y ny se y. Using te distriutive lw to remove prenteses Erlier, we wrote te distriutive lw s Omitting te multiplition signs, we ve Finlly we n rewrite tis s sine multiplition preedes ddition in te order of opertions. Te distriutive lw omes up in mny situtions ut is triky to work wit wen minus signs pper. Te following re te ptterns tt ommonly our. Cution: is red minus sutrt nd is not relted to te produts on te previous line. Note tt. Removing prenteses following minus sign: Te following identity illustrtes very importnt rewriting opertion: removing prenteses tt follow minus sign.. d d

13 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Tis opertion sould e utomti: tink of multiplying te minus sign y e term inside te prenteses. Here re te detils, wi you sould understnd one, ten put side in fvor of te utomti proedure. d Tis is sutrtion prolem. d Apply te definition of sutrtion: u u v u v wit v d d Apply te definition of negtion: v v v wit v d d Use te distriutive lw to multiply y -: d d d d d Remove prenteses. Here is relted emple of te form d d d Tis is sutrtion prolem. d Apply te definition of sutrtion. d Apply te definition of negtion. d Apply te distriutive lw. d Remove prenteses. Eerise..: Fill in te tle elow. Version of distriutive lw Emple wit ; ; ; d??? d? e f g? ut i Wy? Cek te emple y working out ot sides? 8

14 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. j d d k d? l d? m d? n d d o d d Te preeding ws review of ritmeti. Now we move on to lger.. We do lger y rewriting epressions An epression involves numers, vriles, prenteses, nd lger opertions. Some si types of epressions seprted y semiolons re integers: ; 0; ; 9; 8 vriles: ; y; z; ; ; monomils in : 00 0; ; ; ; ; ; polynomils in : 0; ; ; ; ; rtionl polynomils in : ; ; 00 Alger nd first yer lulus fous on polynomils nd rtionl polynomils wit one or two vriles usully nd y, s well s on equtions nd inequlities involving su epressions. Te rdest prt of lger is rewriting epressions orretly. A typil lger prolem is sequene of rewriting opertions, wi n e written using two possile formts: Horizontl: Originl epression Epression Epression Finl Epression Vertil: Originl epression Epression Epression Finl Epression

15 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. It s mu esier to void mistkes wen you use vertil nottion, euse you re opying e epression from te one diretly ove it. You sould tink of tis formt s si prgrp of mtemtis, in wi e epression is sentene. In most ses te finl epression is otined y simplifying te originl epression to mke it esier to understnd. Simplifying numeril epression mens rewriting it s single numer. However, tere re two wys to simplify n epression wit vriles, y rewriting it s simplified sum or s simplified produt. For emple, one you review ow to multiply using FOIL, you ould use vertil formt to rewrite s simplified produt, wit no repeted ftors: or s simplified sum, wit no repeted terms:. Sustituting in epressions Simplifying epressions uses very si rewriting proess lled sustitution. In sketll gme, sustituting plyer A for plyer B mens repling plyer B y plyer A. Similrly, to sustitute epression A for epression B, reple epression B y epression A, wit one tenil requirement: Definition of sustitution: To sustitute epression S for vrile in epression E, erse every letter in E nd reple it y te epression S enlosed in prenteses. Plese engrve te following sttement in te innermost ore of your mtemtil suonsious: Wen you sustitute n epression for vrile, put prenteses round te epression. Here is n informl eerise: try to find tetook tt orretly desries sustitution. Emple..: Sustitute for in te epression Corret Metod: Erse every tere s just one in te epression nd reple it y.

16 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Corret Answer: Inorret Metod: Reple every y Inorret Answer: Two importnt oservtions: Prenteses mke differene, sine 8 weres 9 9 Te nswer is not simplified. It s orret, euse te question sid sustitute rter tn sustitute nd simplify. Sustitution nd simplifition re different proesses: Sustitution is purely menil erse nd reple opertion. Don t tink, just erse nd reple! Simplifition involves rewriting te epression y pplying lger rules. Tinking out lger wen you do sustitution very often leds to errors. Te tle elow lists ommon errors tt our rter frequently. To revel e error, we simplify te result of e orret sustitution. Question mrks pper in te lst olumn wen omitting prenteses produes result wit onseutive minus signs, nottion tt is usully onsidered illegl. Sustitute Corret Inorret for in for in for in for in for in? for in 9 for in 9 for in? for in for in

17 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Te following tle gives emples of sustitution in epressions, one per row: Sustitute For In tis Corret Answer Inorret tis tis f f f f f f f f f Eerise..: Sustitute s indited ut do not simplify in e of te following epressions. for in for in for in d for in e for in f for in g for in for in i for nd for in j k l for nd for in for nd for in for nd for in Eerise..: Even toug we ven t disussed simplifition metods in detil, see if you n rewrite e nswer in Eerise.. s simplified sum. Here is n emple using te nswer to.. i, :

18 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Answer to.. i: Te lws of lger re listed elow. Run troug tem quikly, ten ontinue to te net setion nd refer k to tis list s neessry. Addition lws Identity: 0 Commuttive: Assoitive Bot re written s. Inverse: For e epression tere is n epression wit 0., lled te negtion or negtive of, is equl to te produt. Te ommuttive nd ssoitive lws llow us to ompute sum of tree or more epressions in ny order. Emple: d e e d. All su sums n e written simply s d e. Definition of sutrtion: Te differene of nd is written nd is defined y. Multiplition lws Identity: Commuttive. Assoitive Bot re written s. Inverse: Let e ny nonzero epression. Ten tere is n epression wit te property. In lultor nottion, we write /. 8

19 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Te ommuttive nd ssoitive lws llow us to ompute produt of tree or more epressions in ny order. For emple, d e e d. All su produts n e written simply s de. Definition of division: If is nonzero ten te quotient of y, written In lultor nottion te quotient is / /. If 0, te quotient is undefined. Distriutive lw: te onnetion etween produts nd sums, is defined to e Tis lw is te sis for multiplition rules su s FOIL, wi sys tt d d d. Power lws rules for eponents: ; In lultor nottion: E ^ ^ ^ ^ ^ ^ ^. Te ove lws re lled ioms or definitions. All oter lger rules n e derived s teorems. In prtiulr, you migt wnt to prove te following: Teorem : Eponentil nottion 0 ;, et. d e Teorem : Sutrtion nd ddition re inverse opertions: Teorem : Division nd multiplition re inverse opertions. 9

20 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. In lultor nottion: / / Teorem : Frtion rules: d d d d d In lultor nottion: / / / / d d / d / / d / d. Sustituting in Identities In Cpter, te disussion of lger rules su s te distriutive lw seemed to ssume tt te vriles used were stnd-ins for numers. Te ig piture of lger is tt te vriles in tose lws n stnd for ritrry epressions wi temselves involve vriles. Te min ide is s follows. Te lws of lger, listed in te lst setion, re si identities tt tt pply to ll epressions. Wen you sustitute epressions for vriles in lger lws, you get new identities. Just rememer to put prenteses round te sustituted epressions! Alger involves using wt students usully ll formuls, or rules: we ll ot of tese identities, euse tey re sttements tt two epressions re identil. Given ny epression E, te sttement E E is n ovious identity. Insted of sying tt F E is n identity, we will simply write F E. Wen you do lger, you sustitute for vriles in lger rules to get new identities. Fundmentl Sustitution Priniple: Wen you sustitute n epression for vrile in n identity, te result is n identity. In prtie, tis priniple is pplied s follows: Suppose is vrile, F is n epression you re working wit ontining vrile, nd E is n epression tt you know is equl to tt is, E. If you sustitute epression E for vrile in epression F to get new epression F*, ten F* F. Emple..: Sustitute for, for, nd 0 for in te distriutive lw:. You get new identity: 0 0 Now sustitute for in tis new identity to get tird identity: 0 0 0

21 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Agin we ution tt mny, mny lger errors our euse students forget te prenteses wen tey sustitute. Rememer: Wen you sustitute n epression for vrile, erse nd reple te vrile y te epression written inside prenteses. Eerise..: Sustitute s indited ut do not simplify for nd for in for in for in d for, for, nd for in e for in f for in g for in y y for in i for nd for in j for nd for in k for nd for in l for nd for in m for nd for in n for in f f o for in f p for in f z z Emples n, o, nd p my seem puzzling. Emple n is n identity if f is vrile. In emples o nd p, f is te nme of funtion nd te identity sown is speil kind of identity lled funtion definition, wi will e disussed in te following setion. Eerise..: Find n lger lw wit te following property: Omitting prenteses wen you sustitute for nd for yields flse sttement.. Order of opertions for generl epressions It s triky to work wit epressions tt involve ot vriles nd prenteses. In speil ses su s d e nd de, you n perform te opertions in ny

22 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. order. In most ses, te order of opertions does mtter nd te sfest proedure is to insert prenteses tt indite te order of opertions. Emple..: Write s simplified sum. Notie tt te epression doesn t ve prenteses sowing weter te first opertion is or. However, ording to te order of opertions rules, te multiplition is done first. Now omes te triky prt: Sine 8, you migt egin y writing 8. Tt s not orret! Te order of opertions rule tells you were to insert prenteses: te miguous epression relly mens [ ]. Here we used squre rkets to sow te new prenteses tt were inserted to indite te order of opertions. Terefore, you sould proeed s follows: Te originl epression is Insert prenteses round te produt, sine multiplition omes efore sutrtion. [ ] Find te produt inside te prenteses. [ 8 ] Use te distriutive lw. 8 Collet like terms. 8. To sve step, figure out wi lger opertion needs to e done first nd ten ple prenteses round te result of te opertion, s follows: Te originl epression is Insert prenteses round te produt. [ 8 ] Use te distriutive lw. 8 Collet like terms. 8 Alwys keep in mind te following: lger works y rewriting epressions. You sould Know nd use PEMDAS to follow te order of opertions. Put prenteses round te result of e lger opertion. In some ses, te inserted prenteses migt e superfluous, ut it s etter to e sfe tn sorry! Keep in mind te order of opertions s you red troug te emples elow. For lrity, tey ll sow te inserted prenteses s squre rkets.

23 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. In, prenteses re inserted ot efore nd fter te lger opertion is rried out. You n sfely omit te efore step s in. You my omit te fter step s well, s in, ut e reful! Oter prolems re written out in full detil. 8 8] [ ] [ 8 8] [ 8 0 ] [ ] [ ] [ ] [ ] [ ] [ ] [ [ ] / / / [ ] ] [ ] [ ] [ ] [ ] [ ] [ ] [ 8 9 ] [ ] [ 0 8

24 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. [ ] [ ] [ 8 ] 0 [ ] Eerise..: Rewrite e of te following s simplified sum. d e f g i j k l m n Eerise..: Rewrite e epression in one vrile s stndrd form polynomil, wit no repeted powers nd powers deresing from rigt to left. Rewrite e epression involving two vriles s simplified sum. Rememer to put prenteses round te result of e lger opertion. A] B] C] d e f g

25 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. i j k l y y y y y m n y y. Clssifition of Epressions Tis setion is kind of tenil referene to terminology found in mtemtis tets. It lists te kinds of epressions tt n e sustituted for vriles in te si identities disussed in te previous setion. Some of tis mteril s een overed efore: it s lwys good to review. An integer is wole numer, possily preeded y sign: 0; ± ; ± ; ±,,,. A rtionl numer is quotient of integers: ; ; 8 ; 8 A vrile is letter, usully tken from te Englis or Greek lpet. Te most frequently used vriles re, y, z, s nd t. A positive integer power of is one of te following:,,,,... Tese symols re revitions for repeted multiplition, s follows. 0 ; ;, wi mens times ;.. Negtive integer powers re lternte nottion for reiprols: is wy of writing tt voids using frtions. Anoter point of view: is wy of writing tt voids using negtive eponents. A monomil in is n integer, power of, or n integer times positive integer power of. 0

26 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Te degree of monomil in is te power to wi is rised. It is undefined if te monomil is zero, or is zero if te monomil is n integer. For emple,,,,, re monomils wit degrees 0, 0,,,, nd respetively. Te degree of te zero monomil is undefined. Te oeffiient of monomil is defined seprtely for te tree ses just listed: Te oeffiient of n integer is te integer. Te oeffiient of power is. Te oeffiient of n integer times power of is te integer. Here re nine monomils: 0,,,,,,,, Teir oeffiients re: 0,,,,,,,,. A polynomil is monomil or sum or differene of monomils. Emples re ; ;. Te lst emple is lwys written. 0 0 A stndrd form polynomil is one in wi ll te monomils ve different powers nd te powers of te monomils derese s you red from left to rigt. If polynomil is in stndrd form, ten te leding term is te monomil wit te igest power of. te leding oeffiient is te oeffiient of te leding term. Every polynomil n e rewritten in stndrd form: reorder nd ten ollet terms y using te distriutive lw.. Emple: Te degree of polynomil is te igest power of te monomils in te polynomil otined y rewriting te originl polynomil in stndrd form. Emples: 0 s leding oeffiient 0. s leding oeffiient

27 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. s leding oeffiient. 8 s leding oeffiient. s degree. s degree. in stndrd form is nd so s degree. A moni polynomil is stndrd form polynomil wit leding oeffiient. Mke sure you know te ove definitions well. Te remining ones elow re more tenil: refer k to tem wen neessry. A polynomil epression is uilt out of polynomils y te opertions of dding, sutrting, multiplying nd rising to positive integer power. Every polynomil epression n e rewritten s polynomil y multiplying out to remove prenteses. For emple: A rtionl polynomil is frtion of polynomils, possily preeded y minus sign. Te frtion line mens tt te top numertor is divided y te ottom denomintor. A stndrd form rtionl polynomil is one wose numertor nd denomintor re in stndrd form nd ve positive leding oeffiients. Emple: Rewrite in stndrd form s follows: A polynomil is speil kind of rtionl funtion, sine p p. A rtionl polynomil epression is frtion of polynomil epressions. Emple:. Every rtionl polynomil epression n e rewritten s rtionl polynomil y multiplying out to remove prenteses.

28 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. A generl rtionl epression is uilt out of polynomil epressions y dding, sutrting, or multiplying two rtionl polynomil epressions, or y rising rtionl polynomil epression to positive or negtive integer power. Here is modest emple Every rtionl epression n e rewritten s rtionl polynomil y possily gret del of multiplying out nd working wit frtions to omine everyting into stndrd form rtionl polynomil. You migt try to do tt simplifition for te ove emple. You will quikly e disourged unless you ve ess to omputer lger system. In te ove emple, eponents re restrited to e integers. If tey re llowed to e rtionl numers, ten te epressions re lled lgeri epressions. You know, of ourse, tt rtionl powers n e written wit rdil signs: te si emple is tt / n e rewritten s. 8

29 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. CHAPTER : SIMPLIFYING POLYNOMIAL EXPRESSIONS You re fmilir wit polynomil epressions in single vrile. Su epressions involve numers, te vrile, nd wole numer eponents. Division signs nd frtions re not permitted. Some emples of polynomil epressions, in list seprted y semiolons, re: 0; ; ; ; ; ; ; ; 8 ; In tis setion, we sow ow to use te lws of lger to rewrite ny polynomil epression s polynomil in stndrd wy.. Monomils nd Polynomils A monomil in is numer, wole numer power of, or non-zero numer times positive integer power of. Emples: ; ; ; We will utomtilly rewrite te monomil 0 k s k, sine 0 A polynomil in is sum of simplified monomils in. Emples: 0; ; ; ; ; ; A simplified polynomil in is sum of simplified monomils in in wi no two monomils ve te sme power. Emples: 0; ; ; A stndrd form polynomil in lso lled polynomil in stndrd form is simplified polynomil in in wi te powers of derese s you red from left to rigt. Emples: 0; ; ; 9; Almost everyting we ve studied so fr: te ssoitive, ommuttive, nd distriutive lws, s well s te order of opertions, will e used in te net setion.. Rewriting polynomil epressions in stndrd form. To multiply powers of, dd eponents 9

30 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Emple:. Use te ommuttive nd ssoitive lws for multiplition to rewrite produt of numers nd powers of s simplified monomil Emple: 8. Use te distriutive lw to omine like terms Emple: We ve desried polynomil s sum of monomils. Rememer tt omintion of dditions nd sutrtions is still sum. Tt s euse sutrtion is defined y. Here is n emple, sown wit n intermedite level of detil. Originl epression wit dditions nd sutrtions 9 Sutrtions rewritten s dditions 9 Reorder te dditions ommuttive lw 9 Rewrite some dditions s sutrtions 9. Use te ommuttive nd ssoitive lws for ddition to rewrite polynomil in stndrd form y prentesizing nd reordering terms, nd ten use te distriutive lw to omine like terms. Wen you rerrnge te terms of te polynomil, rememer tt te sign in front of e monomil is prt of te term, nd must e rried long wen te term is moved. Emple: All detils re sown. Try to solve te prolem y leving out s mny intermedite steps s possile! 9 [ ] [ ] [ ] 9 9 [ 9 ] More emples re sown elow. Emple sows detils of reordering, nd omits te prenteses used in te ove emple. Emple omits detils of reordering, nd tt s ow you sould eventully e le to write out su prolems. 0

31 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor Te following emples, preprtion for te net setion illustrte ow te distriutive lw is used to remove prenteses wen monomil is multiplied y polynomil Rewriting produt of sums s sum of produts E of te si lws of lger desried in te setion on numeril epressions pplies s well to epressions involving vriles. All of te lger lws tt you know re te result of using sustitution in te si lger lws. As n emple, te following mteril sows ow to use te distriutive lw to derive te FOIL lw. d d d Strt wit te distriutive lw Sustitute y for to otin y y y On te rigt side, pply te distriutive lw y y y Use ommuttive lw for to reorder te rigt side y y y

32 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Te result is te FOIL lw. It my look more fmilir if you sustitute for, for y, for, d for to otin te identity d d d In turn, you n sustitute ny epressions wtsoever for te vriles,,, d to otin new identities tt re onsequenes of te FOIL lw. Any identity tells you two metods for rewriting epressions. In tis se: You n rewrite d s d d, or You n rewrite d d s d Te first rewriting opertion, lled multiplying out, rewrites produt of sums s sum of produts. Te seond rewriting opertion, lled ftoring, rewrites sum of produts s produt of sums. Te si ide of multiplying out is tt to multiply two sums of vriles, you multiply e vrile in te first sum y e vrile in te seond sum nd dd te resulting produts: y y y y z y z y z y z y z y z y z Te lst line is n importnt speil se of te FOIL lw, sine Te sme ides work if some dditions re repled y sutrtions. Rell tt nd. Ten y y y y y y y z y z y z We repet n erlier wrning: is red minus sutrt nd s noting to do wit te produts Te following identity differene of squres is of prtiulr importne: nd is usully presented s metod for ftoring te rigt side..

33 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Emples inlude 9y y y Emple..: Use similr rule to rewrite polynomil. s stndrd form Solution: lerly follows te pttern of te following identity, in wi we will sustitute for ; for ; for ; d e d e d e for d; nd for e : 9 Eerise..: Convert e produt of polynomils to simplified sum: first multiply out nd ten reorder nd ollet terms. Te lst two epressions were generted y sustitution in te distriutive lw nd sould yield te sme stndrd form polynomil. ] ] ] d] e] f] g] ] i] j] k] l] m] n] o]

34 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. p] q] r] s] 0 t] 0. Ftoring: rewriting sum s produt In te previous setion, we used te distriutive lw to rewrite produts s sums, proess lled multiplying out. Now we reverse diretion nd use te distriutive lw to rewrite sum s produt. Wen you ftor ver n epression, you rewrite it s produt of simpler epressions, wi re lled ftors noun of te originl epression. Te distriutive lw sys tt te produt of nd is identil to te sum of nd. In oter words, you n ftor s follows Sustituting for, for, nd for, gives Multiply on te left side nd remove etr prenteses: However, we ould ftor te rigt side furter y pulling out more times: We wnt to utomte te proess of ftoring sums su s. To do tis, look for te gretest ommon ftor of nd, nmely te lrgest epression in tis se, te igest power of wi is ftor of ot. In te epression, we see tt nd. We pull out from nd otin. Note tt tis is wt you get from te distriutive lw wen you sustitute for, for, nd for. Tenil note: in te following definitions, ll epressions re ssumed to e rtionl polynomil epressions wit integer oeffiients. Definitions involving ftoring: ll epressions re polynomil epressions Epression is ftor of epression E if tere is tird epression su tt E. In tis se, is lso ftor of E. Epression E is prime if its only ftors re,, E, nd E. Emples re, 9,, nd.

35 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Epression is te gretest ommon ftor GCF of epressions E nd F if tere re ftors e of E nd f of F su tt o E e, F f, nd o e nd f don t ve ommon ftor. In su se we n ftor E F s e f, ut e f n t e ftored ny more, t lest y using tis metod. Emple..: Ftor y pulling out te GCF of te terms. Solution: Te terms of te ve no ommon ftor, ut you proly relize tt we n ftor y te FOIL metod. Here s te lnguge tt would e used. Emple..: Ftor ompletely. Solution: Plese mster te emples in te following tle of ftoring prolems of inresing diffiulty. It is followed y tle of relted eerises. Emple.. Rewrite e sum of terms s produt y ftoring out te GCF of its terms. Ftor te epression: 8 Solution sometimes uses sustitution, sown in te olumn to te rigt 8 Finised:,, ve no ommon ftor. Comments goes into te oeffiients,8, goes into 0, 9 nd 0. Ten goes into,9, Pull out te lowest -power, wi is te GCF 9 8 Pull out te GCF of te oeffiients, 9, 8 nd lso pull out lowest -power.

36 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Pull out lowest -power. Ten pull out lowest -power. [ [ 8 ] [9 ] ] Sustitution used ws Prolem rises in Cl s derivtive of Eerise..: Ftor e of te following sums y pulling out te GCF of te terms, or write DNF does not ftor using GCF metod. Ten rewrite e ftor s simplified sum d e 0 0 f g i j k l m n o 0 Ftoring using FOIL is n importnt tenique tt is disussed in Cpter 9. Te identities in te following eerise re importnt emples of te FOIL metod. Eerise..: Rewrite e epression s produt of sums y using one of te following identities

37 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. ; ;. 9 d e y f y g y 9y i j 0 k l y y m y y n y y o 9 p 8 8. Rewriting epressions wit rdils A squre root n e written in two wys, nd. Te defining property of tese epressions is tt, wi ould lso e written in te form. Be wre tt tere is sutlety wen we del wit numers rter tn epressions: te usul onvention is tt ut not, even toug. In te following identities involving nottion for eponents, te letters m nd n n represent ny epression. In lmost ll te emples we ll work wit, m nd n re integers or rtionl numers.

38 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Lws of eponents mn n m m m n m n m Rdil nottion: m m Emples: / / / / / 9 For te time eing, use te following si fts : Eerise..: rewrite te following produts s simplified sums in wi powers of derese s you red from left to rigt. Te first one is done for you d e f g Te following emples sow ow sum of terms involving rdils n e rewritten s produt y ftoring out te lowest power of. Do not ssume tt one form is simpler tn te oter. Plese fous on te instrutions in te left olumn: tey speify ow te nswer sould e written. Be wre tt te emples 8

39 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Pull out lowest -power in Your nswer sould ontin rdil signs, not frtionl powers / / / / / / / Pull out lowest -power in / Write te nswer wit frtionl powers ut no rdil signs. / / / / / / / Pull out lowest -power nd rewrite using only positive powers. Rewrite s single frtion witout using frtionl powers. Esy wy: to rewrite ny non-frtion s frtion wit denomintor e, use te identity e. Ten 9

40 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Rewrite s single frtion witout using frtionl powers.. Hrd wy: Eerise..: rewrite e epression s indited. As one frtion, witout frtionl powers of Witout using frtions oter tn / By ftoring out, wit no frtionl powers in te nswer d By ftor out te lowest power of e Witout negtive powers f Witout negtive powers 0

41 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor.. Rel life funtions CHAPTER : FUNCTIONS Very often, in rel-life situtions, one mesurement depends on noter. Suppose we wnt to use feet of fene to surround retngulr field. Tus te perimeter of te retngle will e feet. Wt re possile dimensions lengt nd widt for te field? Sine te perimeter is twie te widt plus twie te lengt, it follows tt L W, were L nd W re te lengt nd widt of te field. It follows tt L W nd terefore L W. Of ourse, it s importnt to note tt ot te lengt nd te widt re etween 0 nd. In prtiulr, we require 0 < W <, nd te following tle sows few oies of su W. Possile lengt nd widt of retngle wit perimeter Widt W Lengt L W Cek: L W Te tle sows tt if we know W, ten we know L. We sy tt te retngle s lengt depends upon its widt, or tt its lengt is funtion of its widt. Here s summry of wt we did. Emple..: A retngle s widt W feet nd perimeter feet. Wt is te lengt of te retngle? Solution: Let L e te lengt. Solve te eqution L W to otin L W. Answer: Te lengt of te retngle is W. Emple..: I ve ord feet long. I ut off dmged piee tt is D feet long. How long is te remining good piee? Here we will use very si priniple: Te wole is te sum of its prts. In tis se: Wen you ut ord into piees, te lengt of te ord is otined y dding te lengts of te piees

42 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Solution: Cll te lengt of te good piee G. Ten G D. Solve tis eqution to get G D. Answer: Te lengt of te left over piee is D feet.. Funtion definitions: sustituting for te rgument vrile Mny tetooks sow piture of funtion mine, gdget tt swllows numer lled te input nd spits out numer lled te output. Tt s resonle desription of rel-life funtion. In tis setion, we generlize tt desription y llowing not only numers, ut lso generl epressions wit vriles, s inputs nd outputs of funtions. Te equtions G D nd L W from te previous setion epress te sme ide. A new mesurement of interest to us G or L depends on given originl mesurement D or W. In ot ses te reltionsip n e epressed y lling te originl mesurement. Ten we n write Cll te originl mesurement : Arevite tis s New mesurement originl mesurement. New mesurement New On te lst line, te rigt side is n ordinry epression involving vrile. However, te left side is not n ordinry epression. It looks odd euse it s n Englis word New in it. Write N insted of New. Now we ve: N Te left side looks like N times. But N is not vrile, nd tere is no multiplition involved. Te entire sttement N is speil kind of identity lled funtion definition. It sys tt new mesurement, revited y N, depends on n originl mesurement, revited y.. Here s wt you need to know: Te letter N is te nme of funtion. N s noting wtsoever to do wit N times. is te rgument vrile. Bot sides of te funtion definition re referred to s te vlue of N t. Te rigt side of te funtion definition n e ny epression wtsoever: it my or my not inlude n instne of te vrile. Te funtion definition N is speil kind of identity Te Fundmentl Sustitution Priniple remins true: If E is ny epression, ten sustituting E for in te identity N produes new identity N E E, wi n e rewritten s N E E

43 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. We ve een working wit mny different kinds of epressions. E nd every epression turns into single numer wen numers re sustituted for vriles. However: Tere s no su ting s n epression tt produes more tn one output for given input. In prtiulr, tings su s ± nd ± re not epressions nd my not pper on te rigt side of funtion definition. Emple..: Sustitute for in te funtion definition f to find f f g f g Solutions: f f f f g g g f f g g g Te lst emple is interesting, euse te sustituted epression g ould e eiter funtion vlue or simply te produt g times. It doesn t mtter: in eiter se, te sustitution priniple gurntees tt f g g g is n identity. A funtion is kind of lger lw tt s vlid only for te durtion of prtiulr mtemtil disussion, in ontrst to lger lws su s te ommuttive lw, wi is true lwys. Te euty of funtion definition is tt sustituting ny epression wtsoever for te rgument vrile yields new lger lw. A funtion definition is templte for n infinite numer of lger lws desriing te vlues of te funtion t vrious input epressions. For emple, sustituting for in te funtion definition f yields f. Te doule prenteses to te left of te equls sign re unneessry: we write f. However, te prenteses on te rigt of te equls sign re solutely essentil, sine nd re very different epressions!

44 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Any oter epression, su s f eomes g eomes f g n e sustituted for. Ten. Te proess just demonstrted is lled rgument sustitution. It s esy nd fun to do it wit word proessor: just pste te sustituted epression into te funtion definition, reple e ourrene of te rgument vrile, nd insert prenteses! Study refully te tle elow. It ontins some surprises. In prtiulr, notie te lst olumn, in wi every funtion vlue tt ppers s prt of lrger epression is enlosed in squre rkets. Resons will e given in te net setion. Given f defined elow: f f q q Ten f nd f nd f f z sin z sin sin sin f r sin z sin z sin z sin z f nd finlly f f [ ] [ ] [ ] [ ] [sin ] [sin ] [sin z ] [sin z] [ ] [ ]. Sustituting epressions for funtion vlues Te lst setion empsized tt wen you sustitute n epression for te rgument vrile in funtion definition, te sustituted epression must e enlosed in prenteses. For emple: given te funtion definition f, ten f Wen funtion vlue is used in n epression, you lso need to use prenteses wenever you sustitute for te funtion vlue. Tt s euse te symol f eves like vrile, s sown in te following emple. Agin define f.

45 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Emple..: Sustitute for u in te epression. u u Answer: u u Emple..: Define nd sustitute for f f in te epression. f f Solution: Tret etly like te vrile u in te previous emple. f Answer: f f Emple..: Define nd sustitute in te epression f f f Answer: nd so f f f ] [ ] [ In tis emple, te first set of inserted prenteses ere written for lrity s rkets is not needed, ut omitting te seond set is serious error, s seen from te following Eerise.. Simplify ot te given nswer to.. nd te inorret nswer otined y omitting rkets.. Funtion evlution emples Te following emples sow ow to n epression for te rgument of funtion. Tey lso sow ow to simplify te result. Te simplifition proess depends on following te order of opertions refully. Given, find e funtion vlue nd rewrite it s simplified sum. f 8 9 f z z z f ] [ ] [ f ] [9 ] [ f In te following emples, inserted prenteses will e written s res { }. Given simplify f f f

46 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. } { ]} [ { } { ]} [ { } { } { } { } { f f f f Tis nswer is simplified sum. Tis nswer is simplified produt. Given simplify f f f Step : Simplify f f : Step : Multiply y nd simplify: f f f f } { } { ]} [ { ]} [ { ]} [ { ]} [ { } { } { } { } { f f Given, simplify f f f f f ] [ ] [ ] [ ] [ ] [

47 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. In e eerise elow, find nd try to simplify e of te speified funtion vlues. Eerise.. f f f / ; f ; d f ; e f f Eerise.. f f f / ; f ; d f ; e f f Eerise.. f f f / ; f ; d f ; e f f Eerise.. f f f / ; f ; d f ; e f f. Composition of funtions Emple..: A rel life emple wit numer gme. I give you numer. You ve one ne to doule te numer nd one ne to dd ten to te numer. Wi proedure gives te igger nswer: dding 0, ten douling, or douling, ten dding 0? Solution: Cll te strting numer. In tis prolem, tere re two funtions. Te douling funtion is d. Te dding 0 funtion is 0 Strt wit. Douling first gives d. Adding 0 to te result gives d d 0 0 Tis strtegy strts wit, yields 0, nd n e tougt of s funtion: f d 0

48 Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Strt wit. Adding 0 first gives 0 Douling te result gives d d Tis strtegy strts wit, yields 0, nd n e tougt of s funtion: g d 0 Te result g of dding first, ten douling, is lwys 0 more tn te result f of douling first, ten dding. Te funtions g d nd f d re lulted y putting togeter, or omposing, te funtions nd d. We ve just seen tt te order of omposition mtters. In oter words, omposition of funtions is not ommuttive. We just ssigned somewt rndom nme g to te omposite funtion formed y first pplying, ten pplying d. Te offiil nme of tis funtion is d. It is usully enlosed in prenteses wen we write te funtion definition: d d. Similrly, omposing in te reverse order is defined y d d Tke note: Wen omposite funtion is nmed using te omposition opertor, te rigtmost funtion is evluted first. For emple: to find f g, strt wit, net find g, nd lstly find f g Emple..: Buying lumer. I m uying wood tt osts dollrs per foot. A foot long ord s dmged prt wose lengt is feet. I m llowed to ut off te dmged prt nd py only for te good prt tt I tke. Te rge for utting te wood is 0 dollrs. How mu do I py? In tis prolem tere re tree importnt mesurements nd two funtionl reltionsips. Te lengt of te good prt equls minus te lengt of te dmged prt. Te ost of te good prt is times te lengt of te good prt. For emple, if te lengt of te good prt is 8 feet, it osts 8 dollrs. Te rge for utting te wood is n dditionl 0 dollrs nd terefore I py 8 0 dollrs. In generl, if te lengt of te good prt is G, ten I py G 0 G 0 dollrs. Tere re two wys to proeed, e relted to wy of prsing te prolem. Metod. Let e te lengt of te dmged piee. Find formul for C, te mount I must py. Solution : Nme te quntities s follows. Lengt is mesured in feet. Let D e te lengt of te dmged prt. Let G e te lengt of te good prt. Ten G D s efore. 8