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Function 1st Yr

The document discusses various mathematical concepts related to functions, including definitions of domain, range, and types of functions. It also provides examples and solutions to problems involving functions and their properties. The content appears to be fragmented and contains numerous typographical errors, making it difficult to follow the overall structure.
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0% found this document useful (0 votes)
37 views21 pages

Function 1st Yr

The document discusses various mathematical concepts related to functions, including definitions of domain, range, and types of functions. It also provides examples and solutions to problems involving functions and their properties. The content appears to be fragmented and contains numerous typographical errors, making it difficult to follow the overall structure.
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF or read online on Scribd
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vA ine epee Is Comsidosed os kenconat-| b Every element 3 A has” mepgping in Bie ‘dovt(t)- | : 1 ey element nA ba one he | | %). CAMB) Sb amd Cawe ko B= C | Wy FaeHon Isdbled, by Ea) La Eatery lemcHlon Sa» geben j tau entre fae je not a bunction | eet one mendycelation i not a buvetion ( } [a chock aihater Hélouiny elation LA BIS a Function he f Ty-In2s3} 5 Be} 459893 | Ax YAP) AEA SEB ond boot f Sclulon gy, ew : 7 Sein | The te: turrction trom Reto set pis\g™ a) seme clemted ot dewpi DiS, eral at denon dimen 0 Sok pray ted By Te datterent order foi in “eelatey many hase Sere RET Comforale | Bask Comp: aoe) ty falmrte\) |) og hte teal d ta} { Te is bunedion by ca garis Gunentony, i thie \ ala Noy were sel Aol called clormin nlp Line bur dedined by lornel ty & {abaeds (ary) chy dont b) = BBY | Rare” lok a bometion -— 4 Py sef OF Valse Sad-by bunction ag toe “the domaly is Called’ teamge ob a bunctton ine Lo bili’ Sh buncdion =~ y] Ser Bis Calted Cordomain o& a functions, Node. yo we?) __——f | gunction & (%. TAT VU gud ak bd) = Pea 1 bays ata OS Find Wok 123, uhene bea 3044 solution Giten 4 a “FY Fiyearit . FF) = 313) 4 = Bag Sart ¥r3) Sd bind Pl) it En) = at sebul 7 || | | lass gi elad= Pasar Tuc Od Klads)- E15) ) GEL PE FOU= Fel ater bind HCar-4) Had \. lon ek Lm Belteto” * Leagy x nee Fy Cen roy “Cheats, Blais) = CaassHes(aicjti | “GU . 2 OH oatard SAtasy ze eS fee ec = oak ent Uy = adh Wat oe \ Ui EN TH haed “ae OS | 5) ~ oxy | ' 10) = 4 (a) . a2: = (ose wef vO. ea ‘s Foo wey Wd YES4ar yy |, > = Fas) = Ife 5% ' ee a : =~ a poe Now, Fer | “Ww gah we > gas | 2 QLCAt SE FF ~ 44 we S424 a | 1 0 BUSES) aa | = HER AUD = wae) ene | —| Re AO Fr FON ~VaTR=Q then Find 6 ae] a6 a eh Bag ttn . a me THE (a4) 92) Pic tt “eae the BLE U52.),Feraic) Solusion = top EO) = Sduatin a —— Hes) =Seoy Fen) = VQaea é How Gr = GRE re ie de bx) ~ Va ae FO) > Ven ( ty ~ 5 > Vorenti-2 | sc 2) - Klaxt5)= I Gee aan ee STF sees 2) Se MA) 244 Ret SA 10 then bind FET), bISK-Q), £9) BOF) ond 6 (0) Selution % 6, try bly itd tox0 = “Gao Stats) rie tt ttactio ¢ Wypxdac PP, 'Quc) 7 Lt, 5 DOES LSE e HASH WA HD: NOL ON as SOF 3 OOF yea. daft — rs (ea)+10 ye Lea ? w = det tisy-tosio OS KEM SO bay sd teat 10072 +b Gone: area Say bY) = Lt eqrecono eit Bus to BI44~ 364s+ B10 22 Bl - 2483 3\ He, HHI SRY age | J ech) - 34 Gp tsa 1) fe) bone Testi ag BEI, © Mid “Tae +490 4 Real ahead uenetion = i ss Fah Worction Vik Bike Called eal valued) Ql 60> RSE funclen.. ranged sath Eeoge Chel oh OF Cen fBeloly: fox Bite veda Real bunction’ : | \ gunn Se lomin and rary: St beth ane subset OF Heal SA-490 = TAME eet len b & Called acal funcdions | addiray Linboth Side (eee Band dem FOR): \ = SHAY O14, Demain et th tunction’— Z \ ee Cake eee yore DA sch ok x Valeus Oe tohich J EDIE detind & faijor di Sot se & Bik Value ie Called ag domain . as 4 ae < t _ faery.) ee Ban 00h) domain of E(a) = R40 . Sr dantey: (be) j Rall . 2 2 ee ae a Rules ak Kindy domein, obra gumetion 2 | Din Bah) whi? | y + ., | nt Be |) geaeen? Stpenion 02 Pore Finke Vee, va (Exfemnen 1ettession = 0 BAL CD | WM) FO ny Pee: Ly Oe KAD op WELD eT VEefrennion . . in a, Ja-t fwd he tellowiny bumcHon et ipaac pre= ‘ | MSAyhes OH ic i hoa Weis Solita ter tnile value | : ws divs, EK) = R44] adding 2 bath Sicke = i tye ROCHA) = gn dt eort olution ='- Fon inte Value Ua) (a-a)=0 20 H-E>0 Ks 4 A? aed. 33 eee ea ~i (-ayb3) ao 4 poe 3(8-4)(3-2) sla) J deen ba) = Tot J Ufa, bey = ae ay Ke Solution: for Fnienalue, = WE > 0 x cath WS ‘ i (=a) (3) 6 7 rg Oey. T , 2) a) ta Je ce Cray Mee “erg 20 Crary (Ly C413) ©) CX) ae (4-2) (4-3) (Qt9 Qe ” | sted) ? Nw VO) > sta Solutions bye bine Valu Ss dom BOK) = RFR bt) = A Soutti a Gini, value H-4 ¥O Adding Yin both side Arte oy TORY \ dom BY =RSY}, be) = es Het) = | a ae FAA(AA) FO 7 AAD HAO editing Ain bet Ske mhafitis © We a «dam Fa) = REACH i sae EE HS) Buvon’= Fee Give NA Ae -SAS odiey ain loathed Axor Ware 4 tee D ee AE + gone AR PAY) sl a | 5 a 1 i | | 9 ba) © D2) (rh: we Oye) FTG) jaw 2) 0 Goltulione: fob Waenahee “ ST 2) > fi x x) (Ayla (ado eas > pay i) Comper 0.) tne nian eh : | Je dabe Re nm Rte ae He . “Gt =)G) | | ~) by ( a yet. 4)" ae ae iF at aye: y fra) (2-1 wd nix) = MSD Caper |} ste dom bd «Pasay ie) ~ Ms . : ed War Www ODES calling ain toth stile erdeding Jin both Sie save . “ee aS oy. Fos Kinile volun ret a ine be Ys dom HOY 15.3 a | (8a nay p43 7CREQ 200 + | el BH 320 W420! \WL~310 | Salakient~ Fer ey | Akg HeLa i “xe |. x2 Ua". andatiog Ya tooth ; 2 yaaey-qeyie- stasents || dasha egesy Meguah cig xn yet | F a2 3), C-2€4y fa BC n= Be a) ie we0ie39( ce aes = e Poe IEE) er EE? te ae “) 4 ca SBA ears Bye e (209-9 3) Us-3) (5-3) GS = U4) (4) (8) 2) dom HS ~[4¥) [3 9%) at ey) TT deenain ole = a) 4 | anh JG) Oe Djand res pestiwly Henge de ne oo Cope Surkgeh Peg on EG) . sya Hab 5 ted t } ~ val at DMO Has 3c 407 | aA Ris deine shen od raroon adi. | am ~ sent 4 SF cing dhe, chen or 3s 0g x C= Solon ye400 WY doen W(AJEC CH 502) 5 AL by, ot waresy-< at “wn Fees yes ae sik poet 70 « : ata: - GR 5) 78 Coes ese seus? > 8 ge WT « dort ee AaeNe Z| Fed the domoin= ob OZ y bow = VIE | Soldo | fort bivide Velue - 4-nbo : 7 > tat x20 Et). yg 20D, Sy Laan Ge Pa ie eee | 3: oS op se 7 ae a an or up) "s Sar AHN Sate be ae : ' = (aray(e-3)20 < He 3 NERF C4 aie 9 poy 1 ae 4, pq he z le ae ue ‘ | | or dom Hays (cee gage) Pind The domoin ob- Vq- yp otx)= Via Seldtont— |, \ Selullon2— ‘ Fo Hinile. YoLue : Fore. nite velas 1 | 1k 0 | ey Oe al 0 ~blw a i F(a) 2020 | (= (530 : LS LTOn ir | 7 (31 B-9s0 | (ats) (20 < = Me-3. Hog oo 4 PAs Ae . , Geesy aetayh ww" eset ane fey Chas ka) ata)! qe or 4-4 Ath Ate Te 9 |: oa Aeag 2 9-35 Pepe as Fey ay . +16 = aah a s domiPUQ= (3,3) ain bea) Ca FEPUES =) BO see ) Vara ut Soluiom = of ‘Magen a | nae trom binite. Value. Lat bom Sine Vodue, a ze -4>0 -% > Ero D (143) (1-8 Ne =; { i . ( : 2 2. \ | assy" S)-1 (29 Wa, | ag Pages I = as-4 4-4, be 4,) eee 200) Yave bar 4 = 16 aoe ete 8G 8 olen Ga) = (-cos-2y U3) selubion ?— or binite Value AH S0 GJIE>O (ery (s-0 8 fe sacar sides te rsd 2 ase oA 3k wy dom FCM) = 1-599) shee) deve) pore ray “ee Bew” ae asnug jay eb)" accf o\e zed ae a | K-2c>0 at-(of >0 (ass) OT) 30 WET HET t-GPass; 1 ee, 3G - as q-ar =" = rls oy dom Fas Ge s6) uLG0) QS) b= ya Ha Sebadlen = bog Gite value A220 K-320 5 RL KE Ze 2g) S dom bry) = (3, [3..02) i RQ ‘2 71.39%, 4 | Forge The geal Kot 2 or) bunction Y= by -8 FReomeson bor, binding, outreange “ob tumetion ¢ | aoe For the bunedion = bO4) A} God A owe values obs of i GLoriant valued or. with cerfest to 9 a Pilidabte Sebelduli oy’ abl s " fe the. Wanda 10 ellet a ee bre dm iy Coches ot oxplicttly nae nik 1 bee (ike! le ae 7 A4> WL yw = A-ayos "i » pre = Fray)» - - { abs dae : | = Vir ay | ‘ i | i eerues adding P37 ~ Lin both side | Braheteok + AY ESY = AS — rane Nay 4) = O92), 1 oeflikcsy astay “ i 2 ugh dan pRBd-AD fet ow fea | _ Me Piasxzy ae Byte 8 Finke canny? SEAS) ae 1-ag sed aged os \ J remnyeta)= BAA Find the range oF the bunedion KO) = aH gabon peas gna es | a : det, $= DANS Ww = WH AktF may - ARB!) BC 2) 28 weds ’ gat . bon bine ceonge y-2 SFB rromy F(a)= R$ 24 YD) bey =v p | Sebuten o= BO) VIE yp pa bbs Jav 3° aa) L = Meh = te <4 x bot aeihe tome oo ion= BR | bak or bene thet he-domaln atthe Swen reef TT Monge H=TB 1) | Phe) = Va ‘ Selubion— VIS. 22 FOr A OE for biakle domain’ | bak AUF 0 | ing Lin both Sid -3tvd nave! | GP 0-2 '(0) $e | 2 NR ETE “B+ pte t ean) eh pedo fy doko e (290) | | , | 7 KEYS ( i mae = Se TARR eo J rong bay,» (Ore) tO) afte SHAT Fey Firlke deren 1-20 FUS-F 20 7 (AOU 20. = 8 A £4 ay he bash r t2yo a-(0) = id =i-d =-L-y = 33 -3 209g aL = Sido BUA) = EA Hea) = WEat <= To ELA) = Wo-3 Solution 1 AN-Z=0 . Adding £2 in bet GEE. q AnfrfsroF2 WFR re EF dome GTS 9) S> Vans Seyort dts eet Gide i So hes) | + | y Qr=3 i Sis “a ovat SB | | | Sea i" as be we Ineg i aeb tha agen pe bum etions 1g TM Sernain 9100), Corse”, ACCOR Lin) gy >), e, He! Lat BRO BAe oot PE f en] toy =) 3 i Aedety oa den BNE Tf EG Fos VC roy lent (on Loge Pg aa | 2 yoy sah | oven SOF gia t yetty> * 3 O | bm» Be Solution — qe Fe fon finite Tonge ‘ i 4 Baye i 7 bee as-3 Ss a t= BARS : x | AS ety-3, | q | A132 55 ' wosbtiohy a both (de asswtsort a2 0 Gs z0 -4)20 Oy GE i 1 | | 4 -(-0) 9H ge (cL entesomy seg Gelb a ss ebm) = [23,3] meng Hx) = £053] ay i 20 (sy G) 20 (a4) (sx) 20 get de aeyee Gee gO) att ari, adc bh aoe “i ¢ 36 ~ 4 } Ream bX) Ss Soares} aaa Yeoh SR #35 oe E010) rang bl) = Rlebel fo davinite: namge range 06 the. bunction P= =) but ob emo ‘thet Ane doroin ob the Seat tench bunesion is (0 400) $2 be ‘element ie crcéon ding tort” Mapbra ok teal valu bunction Eaualtly ok funetiing Tole bunetion beryl. 9 ae bald Hobe eral Den ere: Leary ay We Viaayt ay S40 . » domain(s) ~ domain( y) vi Co=\-Sormain (4) =c0zdomaingg)). sso wy Fore any iin demain gra)= Jerry t " “ELE FEN PBs Ae MAT BaF akey Qik > B Bees a tay +3 } domain (Ledomenl|) es ey | » Co- dowsing) = Co~dorrain (4). | Koda gew | 7 ss J= 8% bla Par _ FapbeMag otnts ih 4 bse bar] al 4 took BA) iQ) ee IG) 3.309 X35 Ke WL)? 2 (08) Sl@h 6 3 bE) =TE) SS vy ©S Sw enxi- GO) = Solutent— Orgy my Hey + glo: = a (antd) 4 ipa see aan, on |G). Sulled ok hay" ponctions, a] ety be} 1X 7 R towns» r | Addon ob two buncton i— OO Rr rr het, bs 92% > Boston » , | The dolitjon ob--fev0 dem bumertion Ketuen as (ety 1 detingk by ( FEY = HA (1) ,2LE X ' | The, dlomain o- +9 is @yual 40 the nen Mand You ize..Dr 4g = PEND). ae Jon) Ergot: eA bt) aa ee \ The, Quotiemt Ob d(x) and $l4):46 denoted by Golatens—voilsiv) oa) ae deblend by, re te, Kepand 50) 4 (Bip = 600-4 7 DE. ven dy - {aout 5 = x + \ : wo. san banca en BUR GO are _Sibines Hn. ok ante = Nets 90 9 Reithensy Heian 4") elution s— eA), The Sub-Heefion 8b--fd.tonctlen redunen a5), oy "yatta oe cetined by" (gyn HFC, ERs | ae | The dlorraje ob 6-9 ho +654 fo the eresectis y . (£2 Gnge ie) i FEA) ond fa) gt a. | G | Muliplicadion of Seabee queatity ev He Or Pa DiNDy 5 Liplication oF . 7 FADES Some, BAe Hoya Metyeanads one \ | 7 elon suenieeneaast e KA) Soutien (Eig 2 Fagg enim HOO,2 ame | Ka UD = We. , poof Golltion = geene, a ote WE = neon OO retie ot FOAY YE Mulliphicocion ok jue Ganedion + — | ee 90") =V ETAL tind Katy 19 WFR then, { P Hg '2 1b) +91) HX) -4¢ “Th walhittication of v0 SuncHn ref eken asl) Oa tae bY Ji devine’ by (89) 4 = FU GKD- HEX i Nees ; be NTH MURK “he denen ok 9 ic onial FOAL intention |) ED= HOR GY LO) BRE Re FQ) ond SS) \ = Vian- view Vow SST ve Drege DeND, ake S pe tO! ON * fra | = Gnd “He domein i= oe 9 7 FOO Net yy Ee | ee : det, sR ZR. be debined aely by 5O%) = wa for. tinle demote Vows Cal GLO = anna Grd ( Hos ot: Fond a So extrxo Solution — ~ 28 i J= boy sols echt Virco... / bag rig 45 sistan ai] et ‘ = AAAS 5 2 “EE isin) ws | Sons) aihthy {| awd) ) | Wa Geis) Coie a) - x ee ty ~ bt Vey-¢ 5 : | 2 ared)tt Qty 0) B® ‘ | fe Vey ef = ane DRIES 9 a 7 he } ReveD | | = ann 3 Va == | TE CY = bdioge Cie) fae row tt Feperg ee ~ ACHES ve) Leis | = F(a . : . fon dlont(oy a= § ya 4-4 a toluhin 2 Gun ogee By Some reel bunction end hie grep he , 3 > Constant bunedton 7A, bumction k2 —R (DER) is ee i uncon iW i ees & | yA gre Si ooo ae 1s ‘odd Tenclion ont Cake 7 va Fonction bSR- py Rdatined by. 7 ren ome oe nae eet . CMe ANKE PIG Called sly bumetion. | % & temetn UO) tall bo cst ton F 5.0116 dakmed by BRee LOO LT Our} | Fea)'= BEARER. 2 2 Fay bumetion dom f= gh | eK boa 7 aes « Fey ea4 ea Fat jog tia » “ KOD= FH to he lemon FO oe orp Wage k a exe - KED=53 ag | gree | Seon | i ' x (ar) yy * 4 ye era | ; | | | ; 4) the Tal go ent - onl |The Te rans < Seager Sh a Gein which at tewelion BUD) tS Said “ped ont 5 Biinotall fades i a YeQ = 4 7 A temetion 7 Py i | ext RF 8H Ve ta F(a) = Osta Fos HATE. a Tod we Sea Ps amon —noretive Mbager. omd Aeris O2s>- SECA) EROS Ana oreelion F ollt { ee Croat een annzx0 ie Called Polyframial 4 Gps ©: » ON oefion «| om Sipe Pebynomial ob dignee 1 | B Yt : : yi | Re HG = ard 4 art Smee | ot ok i { j Qe) = Ss i os Pf Radtoral bine signs — é Fi “The $ Foay= 20D an | nae re gy tonal “huncrion, i | eRe bY. | Sete ee a i sto th oki 7A. bimction 1s tyemebide uth». QoL Detined “the bonctlon odd or emen. GQ) BA) = SxS Aare BER) = SER )HAE-) = -Bx8- a > Gere) 1 FO) = —b ea ” Se ie Feeation ts odeh i) bey = sat cayenton, Selutione— Fed = 308 cae) = BxS-s tee 2 BLK) SELAH torte tpretton Is-emen | B> tey= eg > The Qragh ak @ edd. bumetion ts Symmokeic with ear plies Oe Fie oO Real “bumedion “hen the: binetions “|Z bey - fa CAA VERS) ee 4 Et (y= = &) a Zag hy. eeld-Fla)= 0 fte) eee Ae ae Be A we : LT Toe Co eee pe OY ow Fen et oow tk iy toy Gay) 32k, BEAT =o), | by EA)= bE to Ae temction is odel- Sea a) fay CE | En) og, ON = boy (670+ Va = boy eS cama] w fey [EO WEL) y AVS > Wey [Re | >) (SRR “6S ase oly COrVard) © OR seit eae mere all rR Ris ned lay = ef (m20 ) sh (L0) non ie alto Krew as slide vale bometion | Joke domain “Re ond mone is By Go] in bellu, Rue. eK [alas SL es~ “The Yrath of y= IKI Comivts ob Awd” caress o8thoun NE 7] The rorya syn ic ‘Sum function S- Te crywm bunedion on P dabined &3 5 ~ fe eet ma = { Te a oY The domain oft RW Ao (hot “Exponential bunedion z— SgnH=0 An emponenfial Ieunction ts dabined by = © FQ) (are, a¢t) KER don bays, pt “raha, Re PY ee. ie i ko (o)| a \\ | . (4 ! (a | | Watts abet lew ~ a | WO) = aM, wyer i j EA? bo) = boyy (2%) i | | P oket We X=0 | 4 Solution = Fog (1-2) WY We aad of sal hoy | ype aK Lbs Lee 9 ace hoe a> of itkazy | enw ; | | SN ; kone a2 5 | pony Hy = 207 Inger bn : ag (egg ee fe Bre tos Reten ce eas [0,02 J=0 Lre2g-£ f (seo igen teat The. beveion shy ag sel aoe eS {ene hey | ae peek RE R[nrine =] Copeeomt TRY PPR | ag Note Te Ce crak bined hig Wndkon wegavey) Dy. conenen, binelbn and box Ht w a odd burchiviy SINCE > B-JOOHREY 2| abOD. set) phvaber) wh Eee Ipunciion Can BO dora Mae na . funciin ord on omen bumation = 4

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