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MODÜL MATEMATİK - 2 FONKSİYONLAR Alt bölümlerin Fonksiyonlar KARMA TEST - 1 Karma Testler EDĜOñNODUñQñL©HULU Modülün sonunda 1. yy 4. f : [ - R) Z [ - R JMFUBONM tüm alt bölümleri –2 f ( x ) = x2 + 6x +GPOLTJZPOVWFSJMJZPS L©HUHQNDUPDWHVWOHU ³ Fonksiyon Kavramı - I t 2 y = g(x) 2x G-1 Y GPOLTJZPOVOVO LVSBM BöBôEBLJMFSJO ³ Fonksiyon Türleri t 10 1 –1 IBOHJTJEJS x y = f(x) A) x + 5 + 3 B) x + 5 - 3 C) - x + 5 - 3 ³ Parçalı Fonksiyon ve Fonksiyonlarla İşlemler t 18 õFLJMEF HSBGJLMFSJ WFSJMFO G WF H GPOLTJZPOMBS D) - x + 5 + 3 E) x - 5 - 3 JÀJO BöBôEBLJMFSEFO IBOHJTJ LFTJOMJLMF EPôSV- ³ Fonksiyonlarda Görüntü t 22 EVS A) ( gof ) ( 3 ) < 0 B) ( fog ) ( 1 ) < 0 ³ Doğrusal Fonksiyonların Grafikleri t 27 C) ( fof ) ( 0 ) > 0 D) ( gog ) ( -1 ) < 0 6ñQñIð©LðĜOH\\LĜ E) ( fog ) ( 2 ) < 0 ³ Mutlak Değer Fonksiyonu ve Grafikleri t 32 %XE¸O¾PGHNL¸UQHN VRUXODUñQ©¸]¾POHULQH ·/÷7&34÷5&:&)\";*3-*, 1. MODÜL '0/,4÷:0/-\"3 www.aydinyayinlari.com.tr DNñOOñWDKWDX\\JXODPDVñQGDQ XODĜDELOLUVLQL] ³ Bileşke ve Ters Fonksiyon t 41 '0/,4÷:0/,\"73\".** \\HUDOñU Fonksiyon ÖRNEK 2 2. 1-WFËSUFOGWFHGPOLTJZPOMBSJ¿JO 5. f ( x ) = 3x + 7 GPOLTJZPOVWFSJMJZPS 4x - 3 ³ Fonksiyonların DTAöNnIMüşümleri t 57 fB f ( x ) = g ( 2x + FõJUMJóJWFSJMJZPS \"WF#CPõLÐNFEFOGBSLMJLJLÐNFPMNBLÐ[F- A #VOBHÌSF 1(f4o4fo4fo2. .4. f4) (423) LBÀUS SF\"LÐNFTJOJOIFSCJSFMFNBOO#LÐNFTJOJO #VOB HÌSF G-1 Y GPOLTJZPOVOVO FöJUJ BöBô- 2011 tane ·1 EBLJMFSEFOIBOHJTJEJS ³ Tek ve Çift FonksiyoCJnS WlFa ZrBMOt[ 68CJS FMFNBOOB FõMFZFO JMJõLJZF \" ·a EBO#ZFUBONMGPOLTJZPOEFOJS ·b ·2 A) g–1( 2x + 2 ) g–1 ^ x h - 2 A) 2 B) 5 C) 13 D) 4 E) 19 <HQL1HVLO6RUXODU ·c ·3 B) 'POLTJZPOMBSHFOFMMJLMFG H IHJCJTFNCPMMFSMF ·d ·4 25 2 2 ³ Karma TestlHeËrTUFSJtMJS 73 ·5 õFLJMEFWFSJMFOGGPOLTJZPOVOVOUBON HÌSÑOUÑWF C) g–1 f x - 2 p D) 2g–1( x ) + 2 <(1m1(6m/6258/$5 2 Fonksiyonlar ³ Yeni NesÖilRSNEoKru1lar t 81 EFôFSLÑNFMFSJOJMJTUFCJÀJNJOEFZB[BMN ¦Ì[ÑN E) g–1^ 1x.h+ 2\"OLBSBhEB UBLTJMFSJO UBLTJNFUSF B¿Mõ ÐDSFUJ 5- \"öBôEBLJ7FOOöFNBMBSJMFWFSJMFOJGBEFMFSEFOIBO- GGPOLTJZPOVOVOUBONLÐNFTJ \"= {B C D E} 2 EJS)FSLNTPOVOEB5-ÐDSFUBMONBLUBES 3. ôFLJMEFLJ HËSTFMEF CJS DJTJN CFMMJ CJS ZÐLTFLMJLUFO GGPOLTJZPOVOVOEFóFSLÐNFTJ #= { } ZVLBSEPóSVBUMZPS HJMFSJGPOLTJZPOEVS GGPOLTJZPOVOVOHËSÐOUÐLÐNFTJ G \" = { } f g h 2Ö. ?RNEK 3 \"OLBSBhEBUBLTJZFCJOFOCJSLJöJOJOHJEJMFONF- AB AB AB ·a ·1 ·a ·1 ·a ·1 TBGF JMF ÌEFEJôJ ÑDSFU BSBTOEBLJ JMJöLJZJ JGBEF 0RG¾O¾QJHQHOLQGH\\RUXP ·b ·2 ·b ·2 ·b ·2 \\DSPDDQDOL]HWPHYE 1. ? 2. ? · c ·3 ·c 1 ·3 · c · 3 1. ? FEFOGPOLTJZPOVOHSBGJôJBöBôEBLJMFSEFOIBO- EHFHULOHUL¸O©HQNXUJXOX VRUXODUD\\HUYHULOPLĜWLU HJTJEJS $\\UñFDPRG¾OVRQXQGD a) b) c) G\"\"3 G Y =Y - 1 ve A = { - }PMEVôVOB 3. gYöCsJSteNrmBMeOkBüMzõerGeJZBfU(Ox)G= Yx 2BA-Z)O4xN+BM1yO0(TTLBfo)UnõkGsJZiyBoUnOu 6. Z=G Y EPôSVTBMCJSGPOLTJZPOPMNBLÑ[FSF HÌSF B) y (TL) ( fof ) ( x ) = 9x - 4 44 UBONMZPS I. G \" HÌSÑOUÑ LÑNFTJOJCVMVOV[ 3 3 PMEVôVOBHÌSF G BöBôEBLJMFSEFOIBOHJTJOF II. GLÑNFTJOJTSBMJLJMJMFSI»MJOEFZB[O[ #VNBMOTBUöOEBOFOB[LBÀMJSBL»SFEJMJS FöJUPMBCJMJS x (km) x (km) O1 A) 3 B) 15 C) 9 O 211 E) 6 C) -9 D) 9 E) 11 D) A) -13 B) -11 42 4 $OW%¸O¾P7HVWOHUL C) y (TL) D) y (TL) TEST - 1 Her alt bölümün 4 5 U TBOJZF TPOSB DJTNJO ZFSEFO ZÐLTFLMJóJOJ WFSFO VRQXQGDRE¸O¾POHLOJLOL 3 4 denklem h_ t i = 30 + 3t - t2 PMNBLUBES WHVWOHU\\HUDOñU 3 5BON %FôFS (ÌSÑOUÑ',PÑONLTFJZMFPSOJ,BWSBN ÖRNEK 4 x (km) 3 73 x (km)4. B 5. C 6#. AVOBHÌSF DJTNJOU=TBOJZFWFU= 3. sani- TANIM 1. D 2. B 3. B 1 O 12 3 O #JS \" LÐNFTJOEFO1.# L\"ÐöNBFôTJEOBFL UJBMFOSENFMOGIGPBOOLH-JTJ ; A ;\"öZBFô CEJSBUGBPOOLNTJW- FEFôF4S.LÑNf :FRMFSAJWRFSJvMFeOgJGB: ERFAMFSREFO E) y (TL) ZFMFSBSBTOEBLJPSUBMBNBEFôJöJNI[LBÀNTO TJZPOV G \" A # JMF HËZTPUOFESJMVJSS \" LÐNFTJOF UB- x + 4 Ia)BCO)HGJyMF=;SJZGP5O2L TGJ ZYP O=CF2xMxJS+U+JS31fri(lixy)o=r. -3x - 7 ve g ( x ) = -x +GPOLTJZPOMBSWF- 5 EJS x+2 4 \"ONEBLOÐ\"NFZTBJ UB#OLNÐNMFCTJSJOAGPF)OyELFT=ôJZFPxSO-3BLÑ1LNTFBTDJBE\"BFO)EJySB= xH2-34Z3 H Y = 4 - xf2( A ) = {2, -1, 5} A) 0 B) 1 C) 4 D) 5 E) 2 UBONMGPOLTJZPOEFOJS PMEVôVOB HÌSF H \" BöBôEBLJMFSEFO IBOHJTJ- 3 33 GGPOLTJZPOV\"LÐNFTJOEFOBMOBDO)CyJS=Y1FM-FN5Bx-2 xE-JS1 C 3 OO#LÐNFTJOEFLJCJSZFMFNBOJMFFõMJZPSJTF E) cy)= x2 +1 x (km) O 12 H43+ Z3 I Y = WDPDPñ\\HQLQHVLOVRUXODUGDQ YFMFNBOOOGBMUOEBLJHËSÐOUÐTÐZFMFNBOES A) { 2, 3, 4 } B) { 3, 4, 5 } C) { -3, 0, 5 } EFOJS#VEVSVNZ=G Y CJ¿JNJOEFHËTUFSJMJS D) { -2, -1, 3 } E) { -2, 2, 4 } G\"A B PMNBLÐ[FSFUBONLÐNFTJOEFLJFMF- ROXĜDQWHVWOHUEXOXQXU NBOMBSO G GPOLTJZPOV BMUOEBLJ HËSÐOUÐMFSJOJO PMVõUVSEVóV LÐNFZF CV GPOLTJZPOVO HÌSÑOUÑ LÑNFTJEFOJSWFG \" JMFHËTUFSJMJS(ËSÐOUÐLÐ- NFTJPSUBLË[FMMJLZËOUFNJJMF I G \" = {G Y Y! A }PMBSBLJGBEFFEJMJS 2. :BLUEFQPTVOEBMJUSF - CFO[JOJCVMVOBOWF 4. \"öBôEBLJMFSEFOLBÀUBOFTJEBJNBEPôSVEVS IFS LN EF - CFO[JO UÑLFUFO CJS PUPNPCJ- I. ( fof )( x ) =YJTFGCJSJNGPOLTJZPOEVS 1. f 2 3. I. {–1, 0, 8, 24}, II. {(-1, 0), (0, -1), (3, 8), (5, 24)} 4. f ve g MJOHJEJMFOYLNZPMBLBSöMLEFQPTVOEBLBMBO 2. f^ x h = 2x - 6 - - x2 + 2x + 48 CFO[JONJLUBSOJGBEFFEFOGPOLTJZPOBöBôEB- ** GPH Y =G Y JTFHCJSJNGPOLTJZPOEVS 5. f: A = ( -1, 3 ] A R, LJMFSEFOIBOHJTJEJS GPOLTJZPOVOVOUBONLÑNFTJOEFLBÀUBOFUBN f ( x ) = x2 - 4x *** GPH Y =H Y JTFGCJSJNGPOLTJZPOEVS TBZCVMVOVS GPOLTJZPOVOVO HÌSÑOUÑ LÑNFTJ BöBôEBLJMFS- A) f_ x i = 60 - x B) f_ x i = 60 - x *7 HPG Y = GPH Y JTFGPHCJSJNGPOLTJZPOEVS A) 4 B) 5 C) 6 D) 7 E) 8 EFOIBOHJTJEJS 20 100 7 GPH Y =H Y JTFHCJSJNGPOLTJZPOEVS C) f_ x i = 60 - x D) f ( x ) = 60 + 100x 5 A) ( -4, 5 ) B) [ -4, 5 ] C) [ -4, -3 ) E) f ( x ) = 60 – 5x \" # $ % & D) [ -4, 3 ] E) [ -4, 5 ) 1. D 2. A 81 3. D 4. B 3. f : A A R, f_ x i = 6x - x2 + x - 2 6. f (x) = x2 - 6x + 9 - x2 - 12x + 36 x-4 GPOLTJZPOVOVO HÌSÑOUÑ LÑNFTJOEF LBÀ UBOF UBNTBZWBSES GPOLTJZPOVOVOFOHFOJöUBONLÑNFTJBöBôEB- A) 5 B) 7 C) 9 LJMFSEFOIBOHJTJEJS D) 11 E) 12 A) [ 0, 6 ] B) ( 0, 6 ) - { 2 } C) [ 0, 6 ) D) ( 0, 6 ] E) [ 0, 6 ] - { 4 } 1. D 2. C 3. E 5 4. B 5. E 6. B
ÜNwİwVwE.ayRdinSyaİyTinlaEri.YcoEm.trHAZIRLIK ·/÷7&34÷5&:&)\";*3-*, MATEMATİK - 2 1. MODÜL FONKSİYONLAR ³ Fonksiyon Kavramı t 2 ³ Fonksiyon Türleri t 10 ³ Parçalı Fonksiyon ve Fonksiyonlarla İşlemler t 18 ³ Fonksiyonlarda Görüntü t 22 ³ Doğrusal Fonksiyonların Grafikleri t 27 ³ Mutlak Değer Fonksiyonu ve Grafikleri t 32 ³ Bileşke ve Ters Fonksiyon t 41 ³ Fonksiyonların Dönüşümleri t 57 ³ Tek ve Çift Fonksiyonlar t 68 ³ Karma Testler t 73 ³ Yeni Nesil Sorular t 81 1. ? 2. ? 1 1. ? 2. ?
·/÷7&34÷5&:&)\";*3-*, 1. MODÜL '0/,4÷:0/-\"3 www.aydinyayinlari.com.tr '0/,4÷:0/,\"73\".* Fonksiyon ÖRNEK 2 TANIM fB A \"WF#CPõLÐNFEFOGBSLMJLJLÐNFPMNBLÐ[F- SF\"LÐNFTJOJOIFSCJSFMFNBOO#LÐNFTJOJO ·1 CJS WF ZBMO[ CJS FMFNBOOB FõMFZFO JMJõLJZF \" ·a EBO#ZFUBONMGPOLTJZPOEFOJS ·b ·2 ·c ·3 'POLTJZPOMBSHFOFMMJLMFG H IHJCJTFNCPMMFSMF ·d ·4 HËTUFSJMJS ·5 ÖRNEK 1 õFLJMEFWFSJMFOGGPOLTJZPOVOVOUBON HÌSÑOUÑWF EFôFSLÑNFMFSJOJMJTUFCJÀJNJOEFZB[BMN \"öBôEBLJ7FOOöFNBMBSJMFWFSJMFOJGBEFMFSEFOIBO- ¦Ì[ÑN HJMFSJGPOLTJZPOEVS GGPOLTJZPOVOVOUBONLÐNFTJ \"= {B C D E} f g h GGPOLTJZPOVOVOEFóFSLÐNFTJ #= { } AB AB AB GGPOLTJZPOVOVOHËSÐOUÐLÐNFTJ G \" = { } ·a ·1 ·a ·1 ·a ·1 ÖRNEK 3 ·b ·2 ·b ·2 ·b ·2 ·c ·3 ·c ·3 ·c ·3 a) b) c) G\"\"3 G Y =Y - 1 ve A = { - }PMEVôVOB B G \"LÑNFTJOEFLJIFSFMFNBO#LÑNFTJOEFLJCJSWF HÌSF ZBMO[CJSFMFNBOMBFöMFöUJSEJôJJÀJOGPOLTJZPOEVS I. G \" HÌSÑOUÑ LÑNFTJOJCVMVOV[ II. GLÑNFTJOJTSBMJLJMJMFSI»MJOEFZB[O[ C H \" LÑNFTJOEFLJ D FMFNBOO # LÑNFTJOEFLJ JLJ FMFNBOMBFöMFöUJSEJôJJÀJOGPOLTJZPOEFôJMEJS *G \" = { -1, 0, 8, 24 } D I \"LÑNFTJOEFLJCFMFNBOO#LÑNFTJOEFLJIFS- II. f = { ( -1, 0 ), ( 0, -1 ), ( 3, 8 ), ( 5, 24 ) } IBOHJ CJS FMFNBOMB FöMFöUJSNFEJôJ JÀJO GPOLTJZPO EFôJMEJS 5BON %FôFS (ÌSÑOUÑ,ÑNFMFSJ ÖRNEK 4 TANIM \"öBôEBUBONWFEFôFSLÑNFMFSJWFSJMFOJGBEFMFSEFO IBOHJMFSJGPOLTJZPOCFMJSUJS #JS \" LÐNFTJOEFO # LÐNFTJOF UBONM G GPOL- a) G;Z2 G Y = x + 3 TJZPOV G \" A # JMF HËTUFSJMJS \" LÐNFTJOF UB- ONLÐNFTJ #LÐNFTJOFEFôFSLÑNFTJEFOJS 2x + 1 \"EBO\"ZBUBONMCJSGPOLTJZPOBLTBDB\"EB C H3Z3 H Y = 3 4 - x2 UBONMGPOLTJZPOEFOJS D H3+ Z3 I Y = x - 1 GGPOLTJZPOV\"LÐNFTJOEFOBMOBOCJSYFMFNB- x2 +1 OO#LÐNFTJOEFLJCJSZFMFNBOJMFFõMJZPSJTF YFMFNBOOOGBMUOEBLJHËSÐOUÐTÐZFMFNBOES a) r ` Z JÀJO x+3 ` 2EVSGGPOLTJZPO EFOJS#VEVSVNZ=G Y CJ¿JNJOEFHËTUFSJMJS 2x + 1 G\"A B PMNBLÐ[FSFUBONLÐNFTJOEFLJFMF- NBOMBSO G GPOLTJZPOV BMUOEBLJ HËSÐOUÐMFSJOJO C r ` 3 JÀJO 3 4 - x2 `3ESHGPOLTJZPO PMVõUVSEVóV LÐNFZF CV GPOLTJZPOVO HÌSÑOUÑ LÑNFTJEFOJSWFG \" JMFHËTUFSJMJS(ËSÐOUÐLÐ- D r ` 3+ JÀJO x-1 NFTJPSUBLË[FMMJLZËOUFNJJMF \" 3ESIGPOLTJZPOEFôJMEJS x + 1 I G \" = {G Y Y! A }PMBSBLJGBEFFEJMJS 11 ²SOFôJOY= JÀJO x - 1 = - b R EJS 22 1. f 2 3. I. {–1, 0, 8, 24}, II. {(-1, 0), (0, -1), (3, 8), (5, 24)} 4. f ve g
www.aydinyayinlari.com.tr '0/,4÷:0/-\"3 1. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 5 %JLFZ%PôSV5FTUJ G\"\"# G Y =Y+G \" = [- ]PMEVóVOBHËSF 7$1,0%m/*m \"LÑNFTJOJCVMVOV[ #JS GPOLTJZPOVO HSBGJóJOEF EÐõFZ EJLFZ EPó- -ãY+ã -ãYã -ãYã SV UFTUJ LVMMBOMBSBL GPOLTJZPOVO Y FLTFOJ Ð[F- SJOEFUBONMPMEVóVIFSCJSOPLUBEBOZFLTFOJ- OF QBSBMFM ¿J[JMFO EPóSVMBS HSBGJóJ ZBMO[DB CJS OPLUBEBLFTFS#VEPóSVMBS HSBGJóJCJSEFOGB[MB OPLUBEBLFTJZPSTBHSBGJLCJSGPOLTJZPOVOHSBGJ- óJEFóJMEJS ÖRNEK 6 ÖRNEK 9 G - Z # G Y = Y - Y + PMEVóVOB HËSF f \"öBôEB WFSJMFO HSBGJLMFSEFO IBOHJMFSJ GPOLTJZPO GPOLTJZPOVOVOHÌSÑOUÑLÑNFTJOJCVMVOV[ HSBGJôJEJS -3 <Y< 2 -4 <Y- 1 < 1 ã Y- 1 )2 < 16 I. y V. y ã Y- 1 )2 +1 < 17 ãY2 -Y+ 2 17 Ox Ox ÖRNEK 7 II. y VI. y Ox Ox G3- { }Z R - { N } III. y G Y = f (x) - 1 GPOLTJZPOVUBONMBOZPS Ox x+k VII. y #VOBHÌSF G EFôFSJOJCVMVOV[ O G Y = - 1 UBONLÑNFTJ3- { 2 }PMEVôVOEBO x x+k-1 Y= JÀJOY+ k - 1 =PMNBMES 2 + k - 1 = 0 k = -1 IV. y VIII. y O 2x 3 -1 -1 G Y = f ( 3) = =-1 O x-2 3-2 x ÖRNEK 8 ZFLTFOJOFQBSBMFMEPôSVMBSÀJ[JMEJôJOEF* ** *** *7WF7* OPMVHSBGJLMFSJCJSEFOGB[MBOPLUBEBLFTUJôJHÌSÑMÑS4B- A = { B C D E } #= { }LÐNFMFSJWFSJMJZPS EFDF7 7** 7***OPMVHSBGJLMFSGPOLTJZPOCFMJSUJS G\"Z#GPOLTJZPOVJÀJOLBÀGBSLMG \" LÑNFTJPMVö- %m/*m UVSVMBCJMJS \" WF # CPõ LÐNFEFO GBSLM CJSFS LÐNF PMNBL a 6 k+ a 6 k + a 6 k+ a 6 k = 15 + 20 + 15 + 6 = 56 Ð[FSF T \" =NWFT # =OJTF\"LÐNFTJOEFO 4 3 2 1 #LÐNFTJOFUBONMGPOLTJZPOTBZTONEJS 5. [–1, 2] 6. [1, 17) 7. –1 8. 56 3 9. 7 7** 7***
·/÷7&34÷5&:&)\";*3-*, 1. MODÜL '0/,4÷:0/-\"3 www.aydinyayinlari.com.tr ÖRNEK 10 ÖRNEK 14 A = { B C D } ve B = { }LÐNFMFSJWFSJMJZPS G3A3 G Y =Y- 1 + #VOBHÌSFG B âPMBDBLöFLJMEF\"LÑNFTJOEFO# JTFG Y+ GPOLTJZPOVOVOG Y GPOLTJZPOVDJOTJO- LÑNFTJOFUBONMLBÀGBSLMGGPOLTJZPOVZB[MBCJMJS deOFöJUJOJCVMVOV[ a !\"JÀJO#LÑNFTJOEFLJEFOGBSLMGBSLMFMFNBO- f (x) = x · 1 +2 G Y+1) = 3Y+1-1 + 2 EBOCJSJJMFGBSLMöFLJMEFC D!\"JÀJO#LÑNFTJOEF- = 3Y + 2 LJGBSLMFMFNBOEBOCJSJJMFFSGBSLMöFLJMEFFöMFOF- 3 CJMFDFôJHÌSÑMÑS =pG Y - 6 + 2 3 · 4 · 4 =GBSLMGGPOLTJZPOVZB[MBCJMJS 3 =G Y - 4 f (x) - 2 = x · 1 3 3 3Y =pG Y -6 ÖRNEK 11 ÖRNEK 15 G3\"3 G Y+ =Y-GPOLTJZPOVWFSJMJZPS G Y mG Ym =Y+WFG = #VOBHÌSFG LBÀUS PMEVôVOBHÌSF G LBÀUS Y+ 1 = 3 Y= 1 Y= 2 f ( 2 ) - f ( 1 ) = 2 · 2 + 5 Y= 1 Y- 2 = 1 Y= 3 f ( 3 ) - f ( 2 ) = 2 · 3 + 5 ÖRNEK 12 + Y= 10 f ( 10 ) - f ( 9 ) = 2 · 10 + 5 f ( 10 ) - f ( 1 ) = 153 (FS¿FMTBZMBSEBUBONM f ( 10 ) = 157 G Y =Y +Y+WFH Y =Y+ GPOLTJZPOMBSWFSJMJZPS ÖRNEK 16 f ( 2k ) = H L FöJUMJôJOJ TBôMBZBO L EFôFSMFSJOJ CV- MVOV[ G Y =YG Y+ WFG = PMEVôVOBHÌSF G LBÀUS ( 2k )2 + 2k + 1 = 3 · ( 3k ) + 3 4k2 - 7k - 2 = 0 Y= 2 f ( 2 ) = 2 · f(3) Y= 3 f ( 3 ) = 3 · f(4) 1 k = – ve k = 2 Y Y= 10 f ( 10 ) = 10 · f(11) f ( 2 ) = 2 · 3 · .... ·10 · f(11) 4 5 f(11) = ÖRNEK 13 10! G3Z3 G Y- =Y +Y+ 1 ÖRNEK 17 GPOLTJZPOVJÀJOG Y+ JGBEFTJOJOFöJUJOJCVMVOV[ G3A3 G Y +Y =Y +Y -YGPOLTJZPOVWFSJMJZPS YAY+ 2 G Y+ 1 ) = Y+ 2 )2 + Y+ 2 ) + 1 #VOBHÌSF G EFôFSJOJCVMVOV[ =Y2 +Y+11 G Y2 +Y = Y2 +Y 2 - Y2 +Y f ( 1 ) = 12 - 1 = 0 10. 48 11. 1 1 13. Y2 Y 4 14. G Y m 5 17. 0 12. - , 2 15. 157 16. 4 10!
'POLTJZPO,BWSBN TEST - 1 1. \"öBôEBLJMFSEFO IBOHJTJ ; A ; ZF CJS GPOLTJ- 4. G3A3WFH3A R ZPOEVS G Y = -Y-WFH Y = -Y+GPOLTJZPOMBSWF- SJMJZPS \" y = x - 1 # y = x + 4 $ y = 5 3 x+2 G \" = { - } % Z= 1 -Y & y = x2 - 4 PMEVôVOB HÌSF H \" BöBôEBLJMFSEFO IBOHJTJ- 4 EJS \" \\ ^ # \\ ^ $ \\- ^ % \\- - ^ & \\- ^ 2. f^ x h = 2x - 6 - - x2 + 2x + 48 5. G\"= - ] A3 GPOLTJZPOVOVOUBONLÑNFTJOEFLBÀUBOFUBN G Y =Y -Y TBZCVMVOVS \" # $ % & GPOLTJZPOVOVO HÌSÑOUÑ LÑNFTJ BöBôEBLJMFS- EFOIBOHJTJEJS \" - # <- > $ <- - % <- > & <- 3. G\"A3 6x - x2 + x - 2 6. f (x) = x2 - 6x + 9 - x2 - 12x + 36 x-4 f_ x i = GPOLTJZPOVOVO HÌSÑOUÑ LÑNFTJOEF LBÀ UBOF UBNTBZWBSES GPOLTJZPOVOVOFOHFOJöUBONLÑNFTJBöBôEB- \" # $ LJMFSEFOIBOHJTJEJS & \" < > # -\\^ $ < % % > & < >-\\^ 1. D 2. C 3. E 5 4. B 5. E 6. B
TEST - 2 'POLTJZPO,BWSBN 1. L!3J¿JO f_ x i = x.3x ve f_ k + 2 i 4. 6f^ x h – [email protected]^ x h + x@ = 2f^ x h + 2x = 15 PMEVôVOBHÌSF LLBÀUS PMEVôVOB HÌSF G Y BöBôEBLJMFSEFO IBOHJTJ f_ k i PMBCJMJS \" # $ % & \" Y- # -Y- $ Y+ % -Y+ & Y- 2. G Y +G Y+ =G Y+ PMNBLÐ[FSF 5. G YZ =G Y +G Z WFSJMJZPS G =WFG = G = PMEVôVOBHÌSF G LBÀUS PMEVôVOBHÌSF G EFôFSJLBÀUS \" # $ % & \" # $ % & 3. G Y =G Y +WFG = 6. GGPOLTJZPOVrY Z! 3J¿JOG Y-Z =G Y -G Z PMEVôVOBHÌSF ff 1 pEFôFSJLBÀUS FõJUMJóJOJTBóMBNBLUBES 4 G =PMEVôVOBHÌSF G LBÀUS \" 2 # 4 % 8 & \" # $ % & 3 3 3 $ 1. C 2. D 3. \" 6 4. C 5. B 6. D
'POLTJZPO,BWSBN TEST - 3 1. f _ x2 - x i = x4 - 2x3 + x2 + 4 I I I I4. B C! R ve B á b J¿JO GPOLTJZPOVWFSJMJZPS ff ax – b p = 50x50 + 49x49 + . . . + 2x2 + x #VOBHÌSF G - LBÀUS bx – a \" # $ % & PMEVôVOBHÌSF G LBÀUS \" # $ % & 2. G3- {} Z3 5. f_ 1– x i = 2x WFG L = -1 ff x + 2 p = x2 + 3 + 4 GPOLTJZPOVWFSJMNJõUJS 5-x x x2 PMEVôVOBHÌSF LLBÀUS #VOBHÌSF G LBÀUS \" - # - $ % & \" - # - $ % & 3. G3- {} Z3 6. G3- {} Z3 f f x - 1 p = x3 - 1 - 7 f_ x i + 3ff 1 p = 1- 2x x x3 x PMEVôVOBHÌSF G LBÀUS PMEVôVOBHÌSF G LBÀUS \" # $ % & \" 1 # 1 $ 3 % 1 & 5 8 4 8 28 1. D 2. C 3. D 7 4. C 5. E 6. C
TEST - 4 'POLTJZPO,BWSBN 1. G3A3CJSFCJSWFËSUFOGPOLTJZPOEVS 4. G3+ A R-UBONMCJSGPOLTJZPOEVS f_ x i = 2.f–1 _ x i - 6 G Y +G Y =Y+Y + PMEVôVOBHÌSF GPG EFôFSJLBÀUS PMEVôVOBHÌSF G LBÀUS \" - # - $ - % - & - \" - # - $ - % - & 2. G\"A B = [ - ]GPOLTJZPOVWFSJMJZPS 5. G3A3 G Y =Y -Y+GPOLTJZPOVWFSJMJZPS A = - ]WFG \" = B G Y =Y- GPOLTJZPOVOVOFOHFOJöUBONLÑNFTJBöBôEB- PMEVôVOBHÌSF #LÑNFTJBöBôEBLJMFSEFOIBO- LJMFSEFOIBOHJTJEJS HJTJEJS \" < > # < $ < \" < > # < > $ <- > % < > & % <- > & <- > 3. f^xh = 1 – 1 6. G Y =Y+ 1 x2 PMEVôVOBHÌSF G Y- GPOLTJZPOVOVO f d x n GPOLTJZPOVOVOFOHFOJöUBONLÑNFTJBöBôEB- 2 LJMFSEFOIBOHJTJEJS UÑSÑOEFOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS \" 3- {} # 3- [ - ] fd x n fd x n f2 c x m \" 2 # 2 $ 2 $ [ - ] % -R - {} 8 4 4 & [ - ] - {} f2c x m f3c x m % 2 & 2 8 4 1. E 2. B 3. E 8 4. \" 5. \" 6. D
'POLTJZPO,BWSBN TEST - 5 1. G3Z3 G Y =Y +Y+ 4. Y y PMEVôVOBHÌSF GGPOLTJZPOVOVOHÌSÑOUÑLÑNF- -1 1 TJBöBôEBLJMFSEFOIBOHJTJEJS - \" < Þ # < Þ $ < Þ B % <- Þ & <- Þ b 2. \"öBôEBLJMFSEFOIBOHJTJ\"EBO#ZFCJSGPOLTJ ôFLJMEFCJSCJMHJTBZBSQSPHSBNOOFLSBOOEBCVMV- yon EFôJMEJS OBOUBCMPWFSJMNJõUJS#VUBCMPEBYJMFHËTUFSJMFOTÐ- UVOEBCJMHJTBZBSBHJSJMFOTBZZ ZJMFHËTUFSJMFOTÐ- UVOEBJTFPTBZOOTGSBPMBOV[BLMóHËTUFSJMNJõ- UJS #VOBHÌSF B-COJOBMBCJMFDFôJFOCÑZÑLEFôFS LBÀUS A) B) C) \" # $ % - & m A BA BA B 5. \"öBôEBLJ HSBGJLMFSEFO IBOHJTJ CJS GPOLTJZPOB ait EFôJMEJS D) A B E) A B A) y B) y xx C) y D) y xx E) y 3. \"öBôEBLJMFSEFO IBOHJTJ CJS GPOLTJZPO EFôJM- x EJS 6. A = % x x = 2n - 1, n ! Z / f: A \" B ve \" R A3 G Y = x + 2 f (x) = x + 1 x2 + 2 2 # N+ A 2 G Y = 2 PMEVôVOB HÌSF G \" LÑNFTJ BöBôE BLJMFSEFO x IBOHJTJEJS $ R A3 G Y = 3 x - 1 % Q+ A3 G Y = 6 x & Q A2 G Y = f 1 x \" 5FLTBZMBS # ¥JGUTBZMBS 3 p $ 1P[JUJGUFLTBZMBS % 1P[JUJG¿JGUTBZMBS & 5BNTBZMBS 1. D 2. D 3. E 9 4. \" 5. C 6. E
·/÷7&34÷5&:&)\";*3-*, 1. MODÜL '0/,4÷:0/-\"3 www.aydinyayinlari.com.tr '0/,4÷:0/5·3-&3÷ #JSF#JS'POLTJZPO ÖRNEK 3 A = {B C D E} ve B = { }LÐNFMFSJWFSJMJZPS TANIM #VOBHÌSF G C =PMBDBLöFLJMEF\"LÑNFTJOEFO# G\"Z#GPOLTJZPOVIFSY1 Y `\"J¿JO LÑNFTJOFUBONMLBÀGBSLMCJSFCJSGGPOLTJZPOVUB- Y1áYJLFOG Y1 áG Y ZBEB ONMBOBCJMJS G Y1 =G Y JLFOY1=YPMVZPSTBGGPOLTJZPOV CJSFCJS - GPOLTJZPOEVS a `\"JÀJOGBSLMEVSVN D`\"JÀJOGBSLMEVSVN ÖRNEK 1 d `\"JÀJOGBSLMEVSVN 4BZNBOOUFNFMJMLFTJHFSFôJ= 24 Af B G GPOLTJZPOV CJSF CJS Br r NJEJS Cr r Dr ²SUFO'POLTJZPO Er r r TANIM ¦Ì[ÑN \"WF#CPõLÐNFEFOGBSLMCJSFSLÐNFPMNBLÐ[FSF rB1 C `\"J¿JO B1áCJLFOG B1 áG C EJS G\"Z#UBONMBOBOGGPOLTJZPOVJ¿JOG \" = B G mGPOLTJZPOEVS EFóFSLÐNFTJOEFLJIFSFMFNBOBLBSõMLUBON LÐNFTJOEFFOB[CJSFMFNBOWBSTB JTFGGPOLTJ- ÖRNEK 2 ZPOVOBÌSUFOGPOLTJZPOEFOJSGËSUFOGPOLTJ- ZPOJTFLTBDBGÌSUFOEJSEFOJS G 3 Z 3 G Y = Y + GPOLTJZPOVOVO CJSF CJS GPOLTJZPOPMVQPMNBEôOBSBöUSBMN G \" á# %FóFSLÐNFTJOEFFOB[CJSFMFNBO B¿LUBLBMZPSTB JTFGGPOLTJZPOVOBJÀJOFGPOL- siyonEFOJS ¦Ì[ÑN Y1 Y `3J¿JOG Y1 =G Y PMTVOG Y1 =G Y j ÖRNEK 4 x21 + 2 = x 2 + 2 & x21 = x22 & x1 = x2 ve x1 =-x2 \"öBôEBLJ öFNB JMF HÌTUFSJMNJö G H WF I GPOLTJZPO- 2 MBSOOCJSFCJSWFÌSUFOZBEBJÀJOFPMNBEVSVNMBS- OJODFMFZJOJ[ PMVS0IºMEFGGPOLTJZPOVCJSFCJSEFóJMEJS %m/*m A f BA g BA hB Br T \" =N T # =OPMNBLÐ[FSF Cr r Br r Br r \"EBO#ZFUBONMBOBCJMFDFLCJSFCJSGPOLTJZPO Dr r Cr r Cr r TBZT P^ n, m h = n! N#OEJS r Dr r Dr ^ n - m h! I. II. III. ±[FMPMBSBLO=NJTF1 O N =OPMVS G mWFÌSUFOGPOLTJZPO ±SOFóJOFMFNBOMCJSLÐNFEFOFMFNBOMCJS H mWFÌSUFOEFôJM JÀJOFGPOLTJZPO LÐNFZFUBONMBOBCJMFDFL-GPOLTJZPOTBZ- I mEFôJMÌSUFOGPOLTJZPO T =EJS 10 3. 24 4. *CJSFCJSWFÌSUFO **CJSFCJSEFôJM JÀJOF ***CJSFCJSEFôJM ÌSUFO
www.aydinyayinlari.com.tr '0/,4÷:0/-\"3 1. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 5 4BCJU'POLTJZPO A = { - - } ve B = { - }LÐNF TANIM MFSJWFSJMJZPS G\"Z# G Y =Y2 -GPOLTJZPOVOVOUÑSÑOÑBSBö- G \" Z # GPOLTJZPOVOEB \" LÐNFTJOJO CÐUÐO USO[ FMFNBOMBS # LÐNFTJOJO ZBMO[ CJS FMFNBO JMF FõMFOJZPSTB G GPOLTJZPOVOB TBCJU GPOLTJZPO f = { ( -2, 3 ), ( -1, 0 ), ( 0, -1 ), ( 1, 0 ), ( 2, 3 ) } EFOJS mEFôJM JÀJOFGPOLTJZPO #BõLBCJSEFZJõMF rY`\"J¿JO G Y =D D`3 JTFGTBCJUGPOLTJZPOEVS ÖRNEK 6 ±SOFóJOG3Z3 G Y = JTF f^ 2 h = 5 G/Z/ G Y =Y+ CJSFCJSWFÌSUFOCJSGPOLTJ f^ 2019 h = 5 f^ x + 3 h = 5 PMVS ZPONVEVS ±[FMPMBSBLD=TF¿JMJSTFG Y =õFLMJOEFLJ mWFJÀJOFGPOLTJZPO GGPOLTJZPOVOBTGSGPOLTJZPOVEFOJS ÖRNEK 7 A = { - } ve B = { }LÐNFMFSJWFSJMJZPS G\"Z# G Y =Y -Y+PMEVóVOBHËSF f fonk- TJZPOVTBCJUGPOLTJZPONVEVS ¦Ì[ÑN G - =G =G =PMEVóVOEBOGTBCJUGPOLTJZPOEVS :BUBZ%PôSV5FTUJ ÖRNEK 8 %m/*m #JS GPOLTJZPOVOVO HSBGJóJOEF GPOLTJZPOVO CJ- G\"Z# G Y = Q+ Y -RY+Q-R+ 1 SFCJSPMVQPMNBEóOBOMBNBLJ¿JOYFLTFOJOF QBSBMFMEPóSVMBS¿J[JMJS:BQMBOCVJõMFNF yatay TBCJUGPOLTJZPOPMEVôVOBHÌSF f ( 20 ) EFôFSJLBÀUS EPôSVUFTUJEFOJS 1BSBMFMEPóSVMBSHSBGJóJFO¿PLCJSOPLUBEBLFTJ- p+1=0 , -2q = 0 jG Y = 0 ZPSTBGPOLTJZPOCJSFCJSEJS p = -1 , q = 0 f(20) = 0 y y = x ÖRNEK 9 x G\"Z# f (x) = 3x - k O 2x + 5 ôFLJMEF G 3 Z 3 G Y = Y GPOLTJZPOVOVO GPOLTJZPOV TBCit fonksiyon oMEVôVOBHÌSF f ( 2) + k HSBGJóJWFSJMNJõUJSYFLTFOJOFQBSBMFM¿J[JMFOIFS UPQMBNLBÀUS EPóSV Z = Y FóSJTJOJ EBJNB CJS OPLUBEB LFTUJ- óJOEFOGGPOLTJZPOVCJSFCJSEJS 3 = - k & k = - 15 & f^ x h = f^ 2 h = 3 25 2 2 f^ 2h+k = 3 15 =-6 - 22 5. CJSFCJSEFôJM JÀJOF 6. CJSFCJS JÀJOF 11 8. 0 9. –6
·/÷7&34÷5&:&)\";*3-*, 1. MODÜL '0/,4÷:0/-\"3 www.aydinyayinlari.com.tr #JSJN²[EFö'POLTJZPO ÖRNEK 12 TANIM G \" Z \" GPOLTJZPOVOEB r Y ` \" J¿JO G Y = Y G3\"3 PMVZPSTB IFS FMFNBOO LFOEJTJOF FõMJZPSTB G Y = B+ b + Y + B- b + Y + B- Y+ b + G GPOLTJZPOVOB CJSJN GPOLTJZPO EFOJS WF I JMF GPOLTJZPOVEPôSVTBMGPOLTJZPOPMEVôVOBHÌSF HËTUFSJMJS f ( 2a -C EFôFSJOJCVMVOV[ ÖRNEK 10 a +C+ 3 = 0 ve a -C+ 1 =PMNBMESB= - C= -1 G Y = -Y+ 1 j f(2a -C = f(-3) =PMBSBLCVMVOVS G;\"; G Y = B+ Y+ b - GPOLTJZPOV CJSJN GPOLTJZPO PMEVôVOB HÌSF B + C ÖRNEK 13 UPQMBNLBÀUS G3\"3 CJSEPóSVTBMGPOLTJZPOPMNBLÐ[FSF G Y = I Y =YPMNBTJÀJOB+ 1 =WFC- 3 =PMNB- G - = G =PMEVôVOBHÌSF G EFôFSJOJCV- MESB+C= 0 + 3 = 3 MVOV[ ÖRNEK 11 G Y =BY+CöFLMJOEFEJS f (-1) = 3 jB C N O L`3 f ( 1 ) = 7 j a +C= 7 j a = C= 5 jG Y =Y+ 5 G Y = N-O Y+N-OCJSJNGPOLTJZ PO f ( 4 ) =PMBSBLCVMVOVS g (x) = n - mx TBCJUGPOLTJZPO ÖRNEK 14 x-k PMEVôVOBHÌSe, m + k JGBEFTJOJOEFôFSJLBÀUS G3\"3 CJSEPóSVTBMGPOLTJZPOPMNBLÐ[FSF G Y- +G Y+ =Y- n PMEVôVOBHÌSF G EFôFSJOJCVMVOV[ GCJSJNGPOLTJZPOJTFN- n =WFN- 2n = 0, G Y =BY+CjG Y- 1) +G Y+ 2) =B Y- 1 ) +C+B Y+ 2) +C=Y- 2 ise HTBCJUGPOLTJZPOJTF -m n = PMNBMES a = C= - G Y =Y- 2 j f(1) =PMBSBLCVMVOVS 1 -k m5 +k= PMBSBLCVMVOVS n 2 %PôSVTBM'POLTJZPO &öJU'POLTJZPOMBS TANIM TANIM B C`3PMNBLÐ[FSF G H\"Z#õFLMJOEFUBONMBOBOGWFHGPOLTJ- ZPOMBS rY`\"J¿JOG Y =H Y PMVZPSTB GJMF G3Z3 G Y =BY+CCJ¿JNJOEFLJGPOLTJZPO- HZFFöJUGPOLJTZPOMBSEFOJSWFG=HJMFHËTUF- MBSB EPôSVTBM GPOLTJZPO EFOJS G CJS EPóSVTBM SJMJS GPOLTJZPOJTFHSBGJóJCJSEPóSVEVS 10. 3 5 12 12. 10 13. 13 14. 0 11. 2
www.aydinyayinlari.com.tr '0/,4÷:0/-\"3 1. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 15 ÖRNEK 19 A = { } #= { } G H\"\"#PMEVóVOBHË- A = { }PMNBLÐ[FSF SF G Y =Y3 + WFH Y =Y2 +GPOLTJZPOMBSFöJU G\"Z\" G G GPOLTJZPOMBSNES öBSUO TBôMBZBO LBÀ GBSLM G m GPOLTJZPOV UBON- MBOBCJMJS ¦Ì[ÑN G =H WFG =H PMEVóVOEBOG=HEJS {(1, 2), (2,1 ), (3, 3)}, {(1, 3), (2,2), (3, 1)} ÖRNEK 16 {(1, 3), (2, 1), (3,2 )} G3\"3 G Y = B- Y + C+ Y -Y+DWF UBOFUBONMBOBCJMJS H3\"3 H Y = B+ Y+ B+C Y + C-D Y+E GPOLTJZPOMBSWFSJMJZPS GWFHGPOLTJZPOMBSFöJUGPOLTJZPOMBSPMEVôVOBHÌSF a +C+D+EJöMFNJOJOTPOVDVOVCVMVOV[ a - 2 = 2a + C+ 1 = a +C -3 =C-DWFD= d denk- ÖRNEK 20 MFNMFSJÀÌ[ÑMEÑôÑOEFB= - C= - D= d = -1 ve A = {YY=L Y# L` N }PMNBLÐ[FSF a +C+D+ d = -PMBSBLCVMVOVS G\"Z# G Y =-YGPOLTJZPOVUBONMBOZPS ÖRNEK 17 GÌSUFOGPOLTJZPOPMEVôVOBHÌSF T # LBÀUS B C`3 f^ x h = 4x2 + bx + 4a - 2 G mWFÌSUFOPMEVôVOEBOT \" = s(B) =PMNBMES ax2 + x + b + 1 GPOLTJZPOVTBCJUGPOLTJZPn oMEVôVOaHÌSF G B+C EFôFSJLBÀUS 4 = b = 4a - 2 & a = b = 2 & f^ x h = 2 & f^ a + b h = 2 a 1 b+1 ÖRNEK 18 ÖRNEK 21 A = { }PMNBLÐ[FSF B C D E`3PMNBLÐ[FSF G BY +CY+D = D- Y + C- Y +DY+E rY1 Y `\"WFY1 YJ¿JOG Y1 G Y PMBDBLõFLJMEF GPOLTJZPOVCJSJNGPOLTJZPOPMEVôVOBHÌSF G C+ d ) G\"Z\"GPOLTJZPOVUBONMBOZPS EFôFSJLBÀUS #VOBHÌSF G +G UPQMBNOOEFôFSJLBÀUS BY2 +CY+D= D- Y3 + C- Y2 +DY+ d GCJSJNGPOLTJZPOPMNBMESG = 2 ve f ( 3 ) = 3 j a = C=D= d = 1 f(2) + f(3) = 2 + 3 = 5 jG C+ d ) = f ( 2 ) = 2 16. –9 17. 2 18. 2 13 19. 3 20. 26 21. 5
TEST - 6 'POLTJZPO5ÑSMFSJ 1. \"WF#CPõLÐNFEFOGBSLMLÐNFMFSPMNBLÐ[FSF 4. f (x) = 4 - x r \" EBO # ZF UBONMBOBCJMFO GPOLTJZPOMBS FMF- cx + 1 NBOMES GPOLTJZPOVTBCJUGPOLsiyoOPMEVôVOBHÌSF r # EFO \" ZB UBONMBOBCJMFO GPOLTJZPOMBS FMF- D+G LBÀUS NBOMES \" 15 # $ 3 CJMHJMFSJWFSJMJZPS 4 2 #VOB HÌSF \" EBO # ZF UBONMBOBCJMFO GPOLTJ- % - 9 & - ZPO TBZT JMF # EFO \" ZB UBONMBOBCJMFO TBCJU 4 GPOLTJZPOTBZTOOGBSLLBÀUS \" # $ % & 2. \"WF#LÑNFMFSJJÀJOT \" æ=T # æ=PMEVôVOB 5. Gå3åZ3 G Y å= Nå+O Yå+åNå+Oå- HÌSF #EFO\"ZBUBONMBOBCJMFOLBÀUBOFÌSUFO CJS JNGPOLTJZPOPMEVôVOBHÌSF Næ+OLBÀ- GPOLTJZPOWBSES US \" # $ % & \" # $ % & 3. Gå3åZå3 6. T \" å=WFT # å= G Y å= B- Yå+å C+ Yå+åBå+åCå- PMEVôVOB HÌSF \"æ EBOæ # ZF LBÀ UBOF CJSF CJS GPOLTJZPOUBONMBOBCJMJS GPOLTJZPOVTBCJUGPOLTJZPOPMEVôVOBHÌSF f^- 2h kaÀUS \" # $ % & \" - # - $ % & 1. C 2. B 3. \" 14 4. \" 5. B 6. D
'POLTJZPO5ÑSMFSJ TEST - 7 1. GCJSJNGPOLTJZPOPMNBLÑ[FSF 4. A = { } #= {B C D E F} G Y- +H Y- =G Y+ LÐNFMFSJWFSJMJZPS PMEVôVOBHÌSF H LBÀUS \" - # - $ - % - & - \"EBO#ZFUBONMBOBOLBÀUBOFTBCJUGPOLTJZPO WBSES \" # $ % & ^ a - 2 hx2 + x - 4 5. f^ x h = ^ a2 - b2 hx2 + ^ a - 3 hx + a - b + 2 2. f(x) = GPOLTJZPOVTBCJUCJSGPOLTJZPOPMEVôVOBHÌSF G LBÀPMBCJMJS 3x2 + 2x + b - 1 GPOLTJZPOVOVOTBCJUGPOLTJZPOPMNBTJÀJO a +CLBÀPMNBMES \" # $ % & \" - # - 7 $ - % - 5 & - 22 3. G GPOLTJZPOV SFFM TBZMBS LÐNFTJ Ð[FSJOEF UBONM 6. T \" =PMNBLÐ[FSF \"EBO#ZFUBONMCJSFCJS WFEPóSVTBMGPOLTJZPOEVS GPOLTJZPOTBZTEJS G =WFG - = - #VOBHÌSF \"EBO#ZFLBÀUBOFTBCJUGPOLTJZPO UBONMBOBCJMJS PMEVôVOBHÌSF G LBÀUS \" # $ % & \" - # $ % & 1. D 2. B 3. D 15 4. \" 5. E 6. E
TEST - 8 'POLTJZPO5ÑSMFSJ 1. Y y 4. G\"Z# GYZY+ 11 H#Z$ HYZY+ 1 B CJ¿JNJOEF UBONMBOBO G WF H GPOLTJZPOMBS CJSF CJS b WFËSUFOEJS D C = {3, 5, 7}PMEVôVOBHÌSF \"LÑNFTJBöBôEB- V Vm LJMFSEFOIBOHJTJEJS :VLBSEBLJUBCMPZBHÌSF ZCBôNMEFôJöLFOJWF \" \\- - ^ # \\- ^ $ \\- ^ YCBôNT[EFôJöLFOJBSBTOEBLJJMJöLJ y =NY+OPMEVôVOBHÌSF a +C+DLBÀUS % \\ ^ & \\ ^ \" # $ % & 2. G\"Z [- R G Y =Y +Y+ 1 ÌSUFOGPOLTJZPOPMEVôVOBHÌSF FOHFOJö\"LÑ- 5. A = { }LÐNFTJWFSJMJZPS NFTJBöBôEBLJMFSEFOIBOHJTJEJS G\"Z\"CJSFCJSWFËSUFOGPOLTJZPOV \" <- R # < R $ -R - % 3 & 3- { -^ G= { B-C C-B } õFLMJOEFUBONMBOZPS #VOBHÌSF B+CLBÀUS \" # $ % & 3. \"öBôEBLJMFSEFOIBOHJTJCJSFCJSGPOLTJZPOBBJU CJSHSBGJLPMBCJMJS A) y B) y Ox Ox C) y D) y Ox O x 6. A = {B C D E} ve B = { }LÐNF- E) y x MFSJWFSJMJZPS O \"EFO#ZFUBONMCJSFCJSGPOLTJZPOMBSEBOLBÀ UBOFTJOJOHÌSÑOUÑLÑNFTJOEFWFFMFNBOMBS CVMVOVS \" # $ % & 1. \" 2. D 3. D 16 4. C 5. C 6. D
'POLTJZPO5ÑSMFSJ TEST - 9 1. T \" =WFT # =PMNBLÐ[FSF 4. ôFLJMEF\"EBO#ZFUBONMBOBOGPOLTJZPOMBSOUÐSMF- #EFO\"ZBLBÀUBOFJÀJOFGPOLTJZPOZB[MBCJMJS SJLBSõMBSOEBHËTUFSJMNJõUJS \" # $ % & AB r\"MJ r\"MJOJOFWJ r7FMJ r7FMJOJOFWJ r$FN r$FNJOFWJ Fonksiyon 5ÑSÑ r \"MJOJO 7FMJOJO FWJOF 7F- #JSFCJSGPOLTJZPO MJOJO$FNJOFWJOF $FNJO \"MJOJOFWJOFHJUNFTJ r \"MJ 7FMJWF$FNJOCJSMJLUF 4BCJUGPOLTJZPO \"MJOJOFWJOFHJUNFTJ 2. G Y = B- Y + B+C Y+ N- r \"MJWF7FMJOJO\"MJOJOFWJOF ±SUFOGPOLTJZPO HJUNFTJ $FNJOLFOEJFWJ- GPOLTJZPOVCJSJNGPOLTJZPOPMEVôVOBHÌSF OFHJUNFTJ a -C+NLBÀUS r )FSLFTJO LFOEJ FWJOF HJU- ö¿JOFGPOLTJZPO NFTJ \" # $ % & #VOMBSEBOLBÀUBOFTJEPôSVEVS \" # $ % & 5. ôFLJMEF Z=G Y EPóSVTBMGPOLTJZPOVWFSJMNJõUJS y 3. G\"Z#CJSFCJSWFËSUFOGPOLTJZPOEVS 6 G Y- =Y+WFSJMJZPS x B = { } PMEVôVOBHÌSF \"LÑNFTJOEFLJFMF- O NBOMBSOUPQMBNLBÀUS y = f(x) \" - # - $ - % - & -1 #VOBHÌSF G +G LBÀUS \" # $ % & 1. D 2. B 3. \" 17 4. C 5. \"
·/÷7&34÷5&:&)\";*3-*, 1. MODÜL '0/,4÷:0/-\"3 www.aydinyayinlari.com.tr 1\"3¦\"-*'0/,4÷:0/7&'0/,4÷:0/-\"3-\"÷õ-&.-&3 1BSÀBM'POLTJZPOMBS ÖRNEK 3 TANIM G3\" R ve rY`3J¿JOG Y =G Y+ PMNBLÐ[FSF 5BON LÐNFTJOJO BZSL BMU LÐNFMFSJOEF GBSLM x + 1, 0# x<2 2# x<5 LVSBMMBSMB CFMJSMFOFO GPOLTJZPOMBSB QBSÀBM UB- f (x) = * x2 - 1, ONM GPOLTJZPOMBS ZB EB QBSÀBM GPOLTJZPO- MBSEFOJS±SOFóJOB C D E!3PMNBLÐ[FSF GPOLTJZPOVUBONMBOZPS #VOBHÌSF G +G UPQMBNOCVMBMN Z g (x), a # x <b ]] b#x#c f (x) = [ h (x), c<x#d ]] \\ r (x), f ( 100 ) = f ( 0 ) = 0 + 1 = 1 f ( 103 ) = f ( 3 ) = 32 - 1 = 8 CJ¿JNJOEF UBONMBOBO G GPOLTJZPOVOB QBSÀB- & f ( 100 ) + f ( 103 ) = 1 + 8 = 9 M GPOLTJZPO H I S GPOLTJZPOMBSOB QBSÀBM GPOLTJZPOVOVOEBMMBSEFOJSB C D EOPLUB- MBSGPOLTJZPOVOLSJUJLOPLUBMBSES ÖRNEK 4 G3\" R ve rY`3PMNBLÐ[FSF ÖRNEK 1 Z 2x - 1, x <-2 ]] x2, 1, f_ x i = * x + 1, x ≥ 1 GPOLTJZPOVWFSJMJZPS f (x) = [ 3x + -2# x #2 ]] x>2 x + 2, x < 1 \\ #VOBHÌSF G + f ( 1 ) + G UPQMBNOCVMBMN CJÀJNJOEFUBONMGGPOLTJZPOVJÀJO G - +G L =G Y=JÀJOG Y =Y+ 2 & f ( 0 ) = 2 FöJUMJôJOJTBôMBZBOLBÀGBSLMLEFôFSJWBSES Y=JÀJOG Y =Y+ 1 & f ( 1 ) = 2 Y=JÀJOG Y =Y+ 1 & f ( 2 ) = 3 f ( -4 ) = -9, f ( 5 ) = 16 & f ( k ) = 25 f(0) + f(1) + f(2) = 7 2k - 1= 25 & k = 13, 13 n (-3, -2) k2 = 25 & k = 5 v k = -5, 5 n [-2, 2] , -5 n [-2, 2] 3k + 1 = 25 & k = 8 ` (2, 3] PMEVôVOEBOLOJOTBEFDF UBOFEFôFSJWBSES ÖRNEK 2 x<1 ÖRNEK 5 x$1 f (x) = * 2x - 2, x- 2, x < 50 4x - 1, f^ f^ x- x $ 50 f (x) = * 10 h h, GPOLTJZPOVOVOUBONWFHÌSÑOUÑLÑNFTJOJCVMVOV[ 'POLTJZPOY<WFYäJÀJOUBONMPMEVôVOEBOUBON GPOLTJZPOVUBONMBOZPS LÑNFTJSFFMTBZMBSES #VOBHÌSF G EFôFSJOJCVMBMN Y< 1 &Ym WFYã 1 &Y- 1 ãPMEVôVOEBO f ( 200 ) = f ( f ( 190 ) ) = .... = f ( ...( f ( 40 ) ... ) HÌSÑOUÑLÑNFTJ -3, 0) , [3, 3 hEVS 17 tane 40 - 17 · 2 = 40 - 34 = 6 1. 7 2. 3 3, 0) , [3, 3) 18 3. 9 4. 1 5. 6
www.aydinyayinlari.com.tr '0/,4÷:0/-\"3 1. MODÜL ·/÷7&34÷5&:&)\";*3-*, 'POLTJZPOMBSEB%ÌSU÷öMFN ÖRNEK 8 G= { - - } ve TANIM H= { - }GPOLTJZPOMBSWFSJMJZPS A 3 3 #3 3 G\"\"3 H#\" R ve A + #áq #VOBHÌSFpG H+ 2, f · g ve f GPOLTJZPOMBSOCV- PMTVO g G+H \"+ B \"3 G+H Y =G Y +H Y MVOV[ G-H \"+ B \"3 G-H Y =G Y -H Y ¦Ì[ÑN GpH \"+ B \"3 GpH Y =G Y pH Y pG= { - - } H+= { - } f f p\"+ B \"3 f f p^ x h = f (x) GpH= { - } ,H Y á g g g (x) f = { - } g D!3PMNBLÐ[FSF DpG Y =DpG Y ÖRNEK 9 öLJGPOLTJZPOVOUPQMBN GBSL ¿BSQNWFCËMÐN- MFSJOJOUBONLÐNFTJ CVJLJGPOLTJZPOVOUBON x - 1, x < 0 LÐNFMFSJOJOLFTJõJNJEJS G;Z; f_ x i = * x + 1, x ≥ 0 GPOLTJZPOVUBONMBOZPS ÖRNEK 6 #VOBHÌSF T ;-G ; LBÀUS G= { - } ve H= { - }GPOLTJZPOMBSWFSJMJZPS f ( Z ) = { ... , -3, -2, 1, 2, ...}PMEVôVOEBO Z - f ( Z ) = {-1, 0} j s (Z - f ( Z ) ) =EJS #VOBHÌSF G+ g ve f -HGPOLTJZPOMBSOCVMVOV[ f + g = { ( -1, 5 ), ( 1, 4 ) }, f - g = { ( -1, -1 ), ( 1, 2 ) } ÖRNEK 7 ÖRNEK 10 G H3\"3 G Y =Y- G-H Y =Y+ %PóBMTBZMBSLÐNFTJOEFUBONMGGPOLTJZPOV 1 - 2x ; x < 20 PMEVôVOBHÌSF HGPOLTJZPOVOVCVMVOV[ f_ x i = * Y+ 2 =pG Y -pH Y Y+ 2 =p Y- 1 ) -pH Y f_ x - 10 i; x ≥ 20 õFLMJOEFUBONMBOZPS x-4 #VOB HÌSF G GPOLTJZPOVOVO HÌSÑOUÑ LÑNFTJOEFLJ H Y = FMFNBOMBSOUPQMBNLBÀUS 3 f ( N ) = {1, 3, 5, ... , 37} 1 + 3 + ... + 37 = 361 x- 4 19 9. 2 10. 361 6. G H { ( –1, 5 ), ( 1, 4 ) }, GmH { ( –1, 2 ), ( 1, –1 ) } 7. 3
TEST - 10 1BSÀBM'POLTJZPOWF'POLTJZPOMBSMB÷öMFNMFS 1. f (x) = * mx - 2, x < 1 Z x2 , x / 0 ^ mod 3 h ]] 2mx + n, x ≥ 1 [ CJ¿JNJOEFCJSGGPOLT JZPOVUBONMBOZPS 4. G3Z R f^ x h = ]] 3x - 1 , x / 1 ^ mod 3 h G - = -WFG = \\ x+1 , x / 2 ^ mod 3 h PMEVôVOBHÌSF G LBÀUS GPOLTJZPOVWFSJMJZPS #VOBHÌSF G - f ( 4 ) +G JGBEFTJOJOEFôF- SJLBÀUS \" # $ % & \" # $ % & 2. 3HFS¿FMTBZMBSLÐNFTJÐ[FSJOEFUBONMCJSGGPOL- 5. 3FFMTBZMBSEBUBONMG Y WFH Y GPOLTJZPOMBS TJZPOV G Y =Y +Y H Y =Y +YPMBSBLUBONMB- OZPS r rY` < J¿JOf_ x i = * x2 - 1 0≤x<4 4 ≤ x < 10 2x – 1 , f^ x h > g^ x h 2x + 4 h^ x h = * x2 + 3 , f^ x h ≤ g^ x h r rY`3J¿JOG Y =G Y+ PMEVôVOBHÌSF I -1 ) +I UPQMBNLBÀUS Ì[FMMJLMFSJOJ TBôMBEôOB HÌSF G + f ( 66 ) \" # $ % - & - LBÀUS \" # $ % & Z 1 , x<0 ]] 6. f^ x h = [ 2 , x = 0 f (x) = * x + 5 x<0 ]] 3. \\ 3 , x>0 3 x -2 x≥0 GPOLTJZPOVUBONMBOZPS õFLMJOEFUBONMZ=G Y GPOLTJZPOVWFSJMJZPS G -Y +Y+ =G Y +Y- G Y =EFOLMFNJOJOLÌLMFSÀBSQNLBÀUS FöJUMJôJOJTBôMBZBOLBÀUBOFYUBNTBZTWBSES \" # $ - % - & - \" # $ % & 1. \" 2. E 3. B 20 4. \" 5. C 6. C
1BSÀBM'POLTJZPOWF'POLTJZPOMBSMB÷öMFNMFS TEST - 11 1. 6ZHVOLPõVMMBSEBUBONMGWFHGPOLTJZPOMBSJ¿JO 4. G Y =Y-WFH Y =Y +GPOLTJZPOMBSWFSJ- G= { - - } MJZPS H= { - - } #VOBHÌSF G+ g ) (- LBÀUS PMEVôVOBHÌSF G-HGPOLTJZPOVBöBôEBLJ- \" - # - $ - % - & -1 MFSEFOIBOHJTJEJS \" { - - } # { - } $ { - } % { - - } & { - - } 5. G3\" A 2. Gå= { - } f^ x h = * 3x + 1 , x # 2 GPOLTJZPOVWFSJMJZPS H= { - }GPOLTJZPOMBSWFSJMJZPS - 2x + 4 , x > 2 Gæ + H GPOLTJZPOV BöBôE BLJMFSEFO IBOHJTJ- y =G Y ÌSUFOGPOLTJZPOPMEVôVOBHÌSF \"LÑ- EJS NFTJBöBôEBLJMFSEFOIBOHJTJEJS \" { - } \" Þ # < Þ $ mÞ # { - } $ { } % mÞ & mÞ > % { } & { } 3. G Y å=Yå+åWFH Y å=åYå+ 6. G3Z3 rY`3J¿JOG Y =G Y+ PMNBLÐ[F- PMEVôVOB HÌSF BöBôEBLJ OPLUBMBSEBO IBOHJTJ SF Gæ+æHGPOLTJZPOVOVOFMFN BOES f_ x i = * x2 - 4 , 0 ≤ x < 4 \" # - $ - 2x - 4 , 4 ≤ x < 10 % - - & GPOLTJZPOVWFSJMJZPS #VOB HÌSF G - G JGBEFTJOJO FöJUJ LBÀUS \" - # - $ - % - & - 1. C 2. \" 3. B 21 4. D 5. E 6. E
·/÷7&34÷5&:&)\";*3-*, 1. MODÜL '0/,4÷:0/-\"3 www.aydinyayinlari.com.tr '0/,4÷:0/-\"3%\"(²3·/5· %m/*m 'POLTJZPO(SBGJLMFSJOJO<FOMFSJ,FTUJôJ /PLUBMBS #JS GPOLTJZPOVO HSBGJóJOEFO Y FLTFOJOF ¿J[J- MFOEJLEPóSVMBSOYFLTFOJOJLFTUJóJOPLUBMBSO %m/*m PMVõUVSEVóV LÐNF GPOLTJZPOVO UBON LÐNF- TJ HSBGJLUFO Z FLTFOJOF ¿J[JMFO EJL EPóSVMBSO #JS G GPOLTJZPOVOVO Y FLTFOJOJ LFTUJóJ OPLUB- ZFLTFOJOJLFTUJóJOPLUBMBSOPMVõUVSEVóVLÐNF MBSCVMVOVSLFOZ=J¿JOG Y =EFOLMFNJOJO GPOLTJZPOVOVOHËSÐOUÐLÐNFTJEJS WBSTB LËLMFSJBSBõUSMS G Y = EFOLMFNJOJO ¿Ë[ÐN LÐNFTJOJO FMF- NBOMBSOBGJOTGSMBSEFOJS ÖRNEK 1 y 5 x (SBGJLYFLTFOJOJLFTNJZPSTBG Y =EFOLMF- NJOJO HFSÀFL TBZMBSEB ÀÌ[ÑNÑ ZPLUVS EF- –6 OJS O m Y=J¿JOZ=G EFóFSJGPOLTJZPOVOZFLTF- OJOJLFTUJóJOPLUBES –7 y (SBGJôJWFSJMFO'POLTJZPOVOUBONWFHÌSÑOUÑLÑNF- y = f(x) TJOJCVMVOV[ 5BON,ÑNFTJ -6, 1 ] , ( 2, 5 ] x (ÌSÑOUÑ,ÑNFTJ[ -7, -1 ) , [ 3, 4 ) m ÖRNEK 2 G GPOLTJZPOV Z FLTFOJOJ Y FLTFOJOJ - WF OPLUBMBSOEBLFTFS ôFLJMEFGGPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS \"ZSDBGGPOLTJZPOVY=OPLUBTOEBYFLTFOJ- y OFUFóFUPMEVóVJ¿JOGGPOLTJZPOVOVOY=OPL- UBTOEB¿JGULBULËLÐWBSESEFOJS f 'POLTJZPOVOHFOFMEFOLMFNJ Bá B` 3 O` ;+ Z=G Y =Bp Y+ Y- O –5 O x m 5 õFLMJOEFJGBEFFEJMJS m #VOB HÌSF G -5 )æ +æ f ( -2 )æ +æ f ( 0 )æ +æ f ( 3 )æ + f ( 5 ) top- MBNLBÀUS f ( -5 ) = 0, f ( -2 ) = 4, f ( 0 ) = 2, f ( 3 ) = -1, f ( 5 ) = 2 f ( -5 ) + f ( -2 ) + f ( 0 ) + f ( 3 ) + f ( 5 ) = 7 1. 5BON, m ] , (2, 5](ÌSÑOUÑ,[–7, –1) , [3, 4) 2. 7 22
www.aydinyayinlari.com.tr '0/,4÷:0/-\"3 1. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 3 ÖRNEK 4 ôFLJMEFZ=G Y GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS y y y = f(x) m x O y = f(x) 5 x – – O 7 õFLJMEF WFSJMFO Z = G Y GPOLTJZPOV JÀJO G Y > 0 FöJUTJ[MJôJOJTBôMBZBOYUBNTBZMBSOOUPQMBNOCV- – MVOV[ – #VOBHÌSF -4 <Y<JÀJOG Y >ES Y= -3, -2, - UBNTBZMBSOOUPQMBN-UJS I. G B =FõJUMJóJOJTBóMBZBOGBSLMBTBZMBSOOUPQMB- NLB¿US 'POLTJZPOVO\"SUBOWF\"[BMBO0MEVôV\"SBMLMBS II. G D = -FõJUMJóJOJTBóMBZBOLB¿GBSLMDHFS¿FMTB- 7ANnM ZTWBSES r Y1 Y ` [ B C] J¿JOY1 >YPMEVóVOEB III. G Y =LFõJUMJóJOJTBóMBZBOGBSLMYEFóFSJPMEVóV- G Y1 > G Y PMVZPSTB G GPOLTJZPOVOB [ B C] OBHËSF LOJOEFóFSBSBMóOFEJS BSBMóOEBBSUBOGPOLTJZPOEVSEFOJS I. f ( a ) = 0 & a = - GBSLMBTBZMBS UPQMBNMB- SEJS rY1 Y ` [ B C] J¿JOY1 >YPMEVóVOEB G Y1 < G Y PMVZPSTB G GPOLTJZPOVOB [ B C] II. y = -EPôSVTVZMBGGPOLTJZPOVOVOHSBGJôJGBSLM OPLUBEBLFTJöJS BSBMóOEBB[BMBOGPOLTJZPOEVSEFOJS III. -2 < k < 4 'POLTJZPOVO1P[JUJGWFZB/FHBUJG%FôFSMFS ÖRNEK 5 \"MEô\"SBMLMBS ôFLJMEFG Y GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS %m/*m y y F y = f(x) a bO c d x – O x – 5 y = f(x) ôFLJMEFLJGGPOLTJZPOV-Þ <Y<B C <Y<DWF #VOBHÌSF E <Y<ÞBSBMLMBSOEBLJIFSYEFóFSJJ¿JOQP[JUJG i) G Y GPOLTJZPOVOVOB[BMBOPMEVôVCÌMHFMFS ii) G Y GPOLTJZPOVOVO BSUBO PMEVôV CÌMHFMFS OF- EFóFSMFSBMNBLUBES GGPOLTJZPOVOVOQP[JUJGPMEVóVBSBMLMBSEBGPOL- EJS TJZPOVOHSBGJóJYFLTFOJOJOÐTUÐOEFEJS i) ( -3, - 2 ] , [ 2, 3 ) B <Y<CWFD <Y<EBSBMLMBSOEBJTFGGPOLTJ- ii) [ -2, 2 ] ZPOVIFSYEFóFSJJ¿JOOFHBUJGEFóFSMFSBMNBLUB- ESGGPOLTJZPOVOVOOFHBUJGPMEVóVBSBMLMBSEB GPOLTJZPOHSBGJóJYFLTFOJOJOBMUOEBES 3. I. 7 II. 4 III. m L 23 4. –5 5. i) ( –3, – 2 ] , [ 2, 3 ) ii) [ -2, 2 ]
·/÷7&34÷5&:&)\";*3-*, 1. MODÜL '0/,4÷:0/-\"3 www.aydinyayinlari.com.tr 'POLTJZPOVO.BLTJNVNWF.JOJNVN%FôFSMFSJ 0SUBMBNB%FôJöJN)[ TANIM %m/*m GGPOLTJZPOVOEBG Y HËSÐOUÐMFSJOJOFOCÐZÐóÐ- y d OF G GPOLTJZPOVOVO NBLTJNVN EFóFSJ CV EF- f(b) y = f(x) óFSJBMEóOPLUBZBJTFNBLTJNVNOPLUBTEF- OJS a f(b) – f(a) GGPOLTJZPOVOEBG Y HËSÐOUÐMFSJOJOFOLпÐóÐ- f(a) x OFGGPOLTJZPOVOVONJOJNVNEFóFSJ CVEFóFSJ b–a BMEóOPLUBZBJTFNJOJNVNOPLUBTEFOJS ab ÖRNEK 6 Z=G Y GPOLTJZPOVOVO[ B C] BSBMóOEBLJPS- y UBMBNB EFóJõJN I[ Z EFóFSMFSJOEFLJ EFóJõJN NJLUBSOO Y EFóFSMFSJOEFLJ EFóJõJN NJLUBSOB 7 PSBOES #V EVSVNEB GPOLTJZPOVO PSUBMBNB 5 y = f(x) EFóJõJNI[ B G B WF C G C OPLUBMBSO- EBOHF¿FOEPóSVOVO E FóJNJPMVS –5 5 x 6 f^ b h - f^ a h m UBOa = m b-a –8 0SUBMBNB EFóJõJN I[OO JõBSFUJ EFóJõJN ZË- [ - ]BSBMóOEBUBONMGGPOLTJZPOVOVO OÐOÐHËTUFSJS0SUBMBNBEFóJõJNI[QP[JUJGJTF EFóJõJN BSUNB ZËOÐOEF OFHBUJG JTF EFóJõJN a) .BLTJNVNWFNJOJNVNPMEVôVOPLUBMBSO B[BMNBZËOÐOEFEFNFLUJS C .BLTJNVNWFNJOJNVNEFôFSMFSJOJCVMVOV[ a) -3, 4 C -8 ÖRNEK 8 ôFLJMEFZ G Y GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS y ÖRNEK 7 8 y = f(x) 5 6 0 x [ - ]BSBMóOEBUBONMGGPOLTJZPOVOVOHSBGJóJõFLJM- EFLJHJCJEJS – O y y = f(x) –6 x #VOBHÌSF BöBôEBWFSJMFOBSBMLMBSEBGPOLTJZPOVO – 6 PSUBMBNBEFôJöJNI[OCVMVOV[ a) [m ] C [ ] D [ ] #VOBHÌSF YG Y #FöJUTJ[MJôJOJTBôMBZBOYUBNTB- d) [m ] e) [ ] f) [m ] ZMBSLBÀUBOFEJS 3 C 3 3 3 f) 0 -3, -2, - PMNBLÑ[FSFUBOFEJS a) – D – d) e) 4 4 10 10 6. B m C m 7. 7 3 33 3 24 8. a) - C D - d) e) f) 0 4 4 10 10
'POLTJZPOMBSEB(ÌSÑOUÑ TEST - 12 1. y 4. y ôFLJMEFG Y 5 y = f(x) GPOLTJZPOVO VOHSBGJóJ WFSJMNJõUJS x x m m O #VOBHÌSFBöBôEBLJMFSE FOIBOHJTJEPôSVEVS \" G3\" 3ËSUFOEJS m # G3\"3CJSFCJSEJS $ GCJSJNGPOLTJZPOEVS :VLBSEB HSBGJôJ WFSJMFO GPOLTJZPOVO UBON WF % G Y =EFOLMFNJOJO¿Ë[ÐNLÐNFTJFMFNBO- HÌSÑOUÑLÑNFTJBöBôEBLJMFSEFOIBOHJTJEJS MES \" [ - ] ,å[ # - , & A = [ Þ JTFG \" = R $ - ,åå[m % [ - , m & [ - , [ - ] 2. ôFLJMEFZ=G Y- GPOLTJZPOVOVOHSBGJóJWFSJMNJõ- 5. ôFLJMEFLJHSBGJLG\"\"#ËSUFOCJSGPOLTJZ POVOHSB- UJS GJóJEJS y y Z G Ym 5 – O x m 6x 8 m G N+ 1 ) =FöJUMJôJOJTBôMBZBONEFôFSMFSJOJO #VOBHÌSF \"-#LÑNFTJOEFLBÀUBOFUBNTBZ UPQMBNLBÀUS WBSE S \" - # - $ - % & \" # $ % & 3. ôFLJMEFZ=G Y GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS 6. ôFLJMEFZ=G Y GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS y y y = f(x) m m O 6 x m x f (x) m m O ≤0 <G[ -1 +G Y ]â x2 + 4 FöJUTJ[MJôJOJ TBôMBZBO GBSLM Y UBN TBZMBSOO FöJUTJ[MJôJOJ TBôMBZBO Y UBN TBZMBSOO UPQMBN LBÀUS UPQMBNLBÀUS \" - # - $ - % & \" - # - $ - % & 1. C 2. B 3. B 25 4. D 5. C 6. \"
TEST - 13 ôFLJMEFLJHSBGJL 'POLTJZPOMBSEB(ÌSÑOUÑ A Z#ZFUBONM 1. y Z=G Y GPOLTJZPOV- 4. ôFLJMEFZ=G Y GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS OVOHSBGJóJEJS y x y = f(x) O m x O #VOB HÌSF \"aG \" LÑNFTJ BöBôEBLJMFSEFO #VOBHÌSF Y- G Y #FöJUTJ[MJôJOJTBôMB- IBOHJTJEJS ZBOYUBNTBZMBSOOUPQMBNLBÀUS \" -RF , { } # -R ={ } \" - # - $ % & $ -R , {} % -R , { } & -R , { } 2. y 5. ôFLJMEFZ=G Y GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS y = f(x) y y=f(x) a m O x m x O 5 ôFLJMEFWFSJMFOGGPOLTJZPOVOBCBóMPMBSBLHGPOL- [ 3, 5 ]BSBMôOEBPSUBMBNBEFôJöJNI[OOPMB- TJZPOVBõBóEBLJHJCJUBONMBONõUS g (x) = * –1 , f (x) ≥ 0 CJMNFTJJÀJOBLBÀPMNBMES –x , f (x) < 0 \" # $ % & #VOBHÌSF H -5 ) + g ( -2 ) + g ( 2 ) + g ( 3 ) top- MBNOOEFôFSJLBÀUS 6. ôFLJMEFZ=G Y GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS \" - # - $ - % & y 6 y=f(x) 3. y –5 O m O x m 7 x m m 5 y = f(x) ôFLJMEF Z=G Y GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS #VOB HÌSF [ -4, 5 ] BSBMôOEBLJ EFôJöJN I[ f (x) JMF[ -2, 0 ]BSBMôOEBLJEFôJöJNI[OOGBSLOO $0 NVUMBLEFôFSJLBÀUS x2 - 16 FöJUTJ[MJôJOJ TBôMBZBO Y UBN TBZMBSOO UPQMBN LBÀUS \" # $ % & \" # $ % & 1. D 2. D 3. D 26 4. E 5. C 6. B
www.aydinyayinlari.com.tr '0/,4÷:0/-\"3 1. MODÜL ·/÷7&34÷5&:&)\";*3-*, %0ó364\"-'0/,4÷:0/-\"3*/(3\"'÷,-&3÷ %m/*m ÖRNEK 2 y ôFLJMEFEPóSVTBMGGPOL- y = f(x) TJZPOVOVO HSBGJóJ WFSJM- B C`3WFG3Z3PMNBLÐ[FSF –1 NJõUJS 2 G Y =BY+CGPOLTJZPOVOVOHSBGJóJEJLLPPS- #VOBHÌSF G EFôF- EJOBUTJTUFNJOEFZ=BY+CEPóSVTVOVOHSBGJ- x SJOJCVMBMN óJOJCFMJSUJS#JSEPóSVOVOHSBGJóJOJEJLLPPSEJOBU O TJTUFNJOEF¿J[NFLJ¿JOCVEPóSVOVOHF¿UJóJFO B[OPLUBZBJIUJZB¿WBSES G Y =BY+C f^ - 1 h = 0 & - a + b = 0 f^ x h = 2x + 2 %PMBZTZMBZ=BY+CEFOLMFNJOJTBóMBZBOFO B[UBOF Y Z TSBMJLJMJTJTF¿JMJQCVTSBMJLJ- 4 MJMFSEJLLPPSEJOBUTJTUFNJOEFJõBSFUMFOJSWFJõB- f^ 0 h = 2 & b = 2 & a = 2 f^ 2 h = 6 SFUMFOFOOPLUBMBSCJSEPóSVQBS¿BTPMVõUVSBDBL õFLJMEFCJSMFõUJSJMJQEPóSV¿J[JMJS ÖRNEK 3 ±SOFóJO G3Z3 G Y =Y+GPOLTJZPOV- OVOHSBGJóJOJ¿J[FMJN x 0 –1 y y = f(x) ôFLJMEF EPóSVTBM G GPOLTJZP- y = f(x) 10 3 x OVOVOHSBGJóJWFSJMNJõUJS y O #VOBHÌSF G B = 2f ( 2 - a ) f(x) = x + 1 1 FöJUMJôJOJ TBôMBZBO B EFôF- SJOJCVMBMN –1 1 x f ( 0 ) = 0, f ( 1 ) = 3 jG Y =Y f ( a ) = 2f ( 2 - a ) 4 3.a = 2. 3 ( 2 - a ) j a = 3 ÖRNEK 1 ÖRNEK 4 ôFLJMEF EPóSVTBM G GPOLTJZP- OVOVOHSBGJóJWFSJMNJõUJS 3FFMTBZMBSEBUBONMGEPóSVTBMGPOLTJZPOVOEB y G - =WFG = -UÐS 1 #VOB HÌSF GPOLTJZPOVOVO LVSBMO CVMVQ HSBGJôJOJ O2 ÀJ[FMJN y G Y =BY+C x GGPOLTJZPOVOVOHSBGJôJÑ[F- y = f(x) SJOEF FLTFOMFSF FöJU V[BL- 1 f ( -2 ) = 1 j -2a +C= 1 MLUBCVMVOBOOPLUBMBSUFT- –2 1x QJUFEFMJN f ( 1 ) = -3 j a +C= -3 –3 j f^ x h =- 4 x - 5 f ( 2 ) = 0 , f ( 0 ) = 1 jG Y = 2 - x 2 33 2 - x =Yj d 2 , 2 n 2 33 2-x = -Yj (-2, 2) 2 1. f^ x h = - 4 x - 5 27 2. 6 4 4. (–2, 2), d 2 , 2 n 33 3. 3 33
·/÷7&34÷5&:&)\";*3-*, 1. MODÜL '0/,4÷:0/-\"3 www.aydinyayinlari.com.tr ÖRNEK 5 %m/*m ôFLJMEFTBCJUI[MBSMBIBSFLFUFEFO\"WF#BSB¿MBSOO Z=BY+CEPóSVTVOVOFóJNJBES ZPMm[BNBOHSBGJóJWFSJMNJõUJS B>JTFZ=G Y =BY+CGPOLTJZPOVBSUBO B<JTFZ=G Y =BY+CGPOLTJZPOVB[BMBO- Yol (km) B ES 20 A 16 Zaman (saat) 1BSÀBM5BONM'POLTJZPOMBSO(SBGJLMFSJ 0 24 TANIM #VOBHÌSF CVBSBÀMBSBSBTOEBLJNFTBGFLBÀODTB- Z g (x), x < x0 BUUFLNPMVS ] ] f (x) = [ h (x), x0 # x < x1 ] ] t (x), x $ x1 \"ZG Y =Y \\ B ZG Y =Y Y-Y= 30 jY= 5 bJ¿JNJOEFLJGPOLTJZPOMBSES Y Y1LSJUJLOPLUBMBSES ,SJUJLOPLUBTBZTTPOMVTBZEBZBEBTPOTV[ TBZEBPMBCJMJS ÖRNEK 6 (SBùL¿J[JMJSLFOLSJUJLOPLUBMBSBSBTBZSBZS ¿J[JMJS ôFLJMEFLJ HSBGJLUF BZO BOEB EJLJMFO \" WF # GJEBOMBSOO CPZMBSOEBLJBSUõHËTUFSJMNJõUJS y (Boy cm) A B ÖRNEK 7 k 30 x , x$0 f_ x i = * -x - 1 , x<0 10 x GPOLTJZPOVOVO HSBGJôJOJ ÀJ[FSFL HÌSÑOUÑ LÑNFTJOJ 02 Zaman (ay) CVMBMN #VOBHÌSF LBÀODBZEBCJULJMFSJOCPZMBSGBSLDN y (- ß PMVS x –1 y (Boy cm) –1 30 20 3x 5 Zaman (ay) 2 0 5. 5 6. 5 28 7. (–1, Þ)
www.aydinyayinlari.com.tr '0/,4÷:0/-\"3 1. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 8 ÖRNEK 11 Z -1 , x <-2 G3Z\"ZBUBONM ]] x + 1; x < 1 f_ x i = [ x + 1 , -2 # x # 2 ]] f_ x i = * \\ 3 , x>2 - 2x + 3; x ≥ 1 GPOLTJZPOVOVOHSBGJôJOJÀJ[FSFLCJSFCJSWFÌSUFOMJôJ- GPOLTJZPOVÌSUFOPMEVôVOBHÌSF FOHFOJö\"LÑNF- TJOJCVMBMN OJJODFMFZFMJN Y< 1 jY+ 1 < Yäj -Y+ãj\"= (-Þ, 2) y [-1, 3]EBÌSUFO 3 (- EB CJSF –2 CJSEJS –1 2 x ÖRNEK 9 ÖRNEK 12 Z x2 - 1 ; x≠1 L`3PMNBLÐ[FSF G;Z; G Y =LY+L mWFËS- ]] ; x=1 UFOGPOLTJZPOVUBONMBOZPS f_ x i = [ #VOBHÌSF G L OJOFOCÑZÑLEFôFSJLBÀUS ]] x-1 \\ 3 mWFÌSUFOPMNBTJÀJOFôJNJOWFZBmPMNBTHF- SFLJS GPOLTJZPOVOVOHSBGJôJOJÀJ[FMJN k = 1 jG Y =Y+ 1 j f ( k ) = f ( 1 ) = 2 k = -1 jG Y = -Y- 1 j f ( k ) = f(-1) = 0 ¦Ì[ÑN G L OOFOCÑZÑLEFôFSJEJS y y = f(x) 3 x 2 –1 1 ÖRNEK 10 y ÖRNEK 13 3 y ôFLJMEF Z = G Y EPóSVTBM GPOLTJZPOV JMF FLTFOMFS BSB- y = f(x) y = g(x) TOEBLBMBOCËMHFZFUBOFLBSFZFSMFõUJSJMJZPS x x y O O 9 õFLJMEFWFSJMFOGWFHGPOLTJZPOMBSOOUBONLÑNFMF- AC SJOJ HÌSÑOUÑ LÑNFMFSJOJ BSUBO B[BMBO PMEVôV CÌMHF- FE MFSJCFMJSMFZFMJN 18 x O BD y = f(x) #VOBHÌSF LBSFMFSJOBMBOMBSUPQMBNLBÀCS2EJS f: T.K =3- {0} g: T.K =3 x y = 1& y = f^ x h = 18 - x , C(a, a) ise a = 6 (,=3- {0} + 3- {0}EBBSUBO (,= (-ß ] 18 9 2 -ß ]EBBSUBO [ ß B[BMBO E(6 + k, k) ise k = 4 j\"MBOMBSUPQMBN+ 16 = 52 8. <m >UFÌSUFO m EFCJSFCJSEJS 29 11. (–Þ, 2) 12. 2 13. 52 10. f: TK =3- {0} (,=3- {0} ve g: T.K =3 (,= (-Þ, 3]
TEST - 14 %PôSVTBM'POLTJZPOMBSO(SBGJLMFSJ 1. ôFLJMEFG3Z3Z=G Y+ GPOLTJZPOVOVOHSB- 4. \"öBôEBLJ GPOLTJZPOMBSEBO LBÀ UBOFTJ 3 Z 3 GJóJWFSJMJZPS ÌSUFOEJS y I. y II. y y = f(x+1) 4 2 O2 x O x y –2 x III. y IV. O 3 G Y- 1 ) =PMEVôVOBHÌSF YEFôFSJLBÀUS x x O O \" # $ - % - & - V. y 2. y y = 2x 2 x O2 6 D C \" # $ % & B6 O x Ax I I ôFLJMEF AB =YCSEJS 5. ôFLJMEFZå=åG Y JOHSBGJóJWFSJMNJõUJS GYZ i\"#$%EJLEËSUHFOJOJOBMBOuGPOLTJZPOVO y LVSBM BöBôEBLJMFSEFOIBOHJTJEJS –3 8x O24 \" Y-Y # –2x2 + 12x $ YmY 3 y = f(x) % Y -Y G Y $ FöJUTJ[MJôJOJ TBôMBZBO Y UBN TBZMBS OOUPQMBNLBÀUS & 2x2 - 12x \" # $ % & 5 3. ôFLJMEFZ=G Y GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS 6. ôFLJMEFG3Z3PMNBLÐ[FSF Z=G Y+ GPOL- y TJZPOV¿J[JMNJõUJS 4 y y = f(x) y = f(x+2) 3 1 –4 x –2 4 6x –5 –3 –2 O 1 2 3 –4 O –2 –3 #VOBHÌSF G G Y =EFOLMFNJOJOÀÌ[ÑNLÑ- G Y- 1 ) = -FöJUMJôJOJTBôMBZBOYEFôFSMFSJOJO NFTJLBÀFMFNBOMES UPQMBNLBÀUS \" # $ % & \" # $ % & 1. \" 2. B 3. E 30 4. B 5. E 6. E
%PôSVTBM'POLTJZPOMBSO(SBGJLMFSJ TEST - 15 1. 3FFMTBZMBSEBUBONM 3. ôFLJMEFZ=G Y WFZ=H Y GPOLTJZPOMBSOOHSB- f_ x i = * - x , x < 0 GJLMFSJWFSJMNJõUJS 2x + 2 , x ≥ 0 y y fonksiyPOVOVOHSBGJôJBöBôEBLJMFSEFOIBOHJTJ- y = g(x) EJS y = f(x) 4 8 A) y B) y –2 x –2 x O O 2 2 x #VOBHÌSF Z=H Y GPOLTJZPOVOVOFöJUJBöBô- x EBLJMFSEFOIBOHJTJEJS –1 O O C) y D) y \" Z=G Y # Z=G Y $ Z=G Y 2 2 % Z=G Y & Z=G Y O x 1 x –1 O E) y x 2 O 4. ôFLJMEFZ=G Y EPóSVTBMGPOLTJZPOVOVOHSBGJóJWF- SJMNJõUJS y 2. f (x) = *–x + 1 , x ≥ 0 GPOLTJZPOVWFSJMJZPS 1 2 x O y = f(x) x , x<0 | |#VOBHÌSF H Y = G Y +GPOLTJZPOVOVO HSBGJôJBöBôEBLJMFSEFOIBOHJTJPMBCJMJS A) y B) y #VOBHÌSF Y+ G Y >FöJUTJ[MJôJOJTBôMB- 3 ZBOLBÀGBSLMUBNTBZEFôFSJWBSES 2 –2 O 2 \" # $ % & C) y x O2 x 3 D) y x 2 x 2 5. Y-Z+ 1 = O1 y 1 EPôSVTVÑ[FSJOEFCVMVOBO\"WF#OPLUBMBSOO E) O1 BQTJTMFSJ GBSL CS PMEVôVOB HÌSF PSEJOBUMBS GBSLOOQP[JUJGEFôFSJLBÀUS 2 \" # $ % & O2 x 1. \" 2. C 31 3. D 4. C 5. B
·/÷7&34÷5&:&)\";*3-*, 1. MODÜL '0/,4÷:0/-\"3 www.aydinyayinlari.com.tr .65-\",%&ó&3'0/,4÷:0/67&(3\"'÷,-&3÷ TANIM ÖRNEK 1 G3Z3 Z=G Y GPOLTJZPOVWFSJMTJO y ôFLJMEFZ=G Y =Y - GPOLTJZPOVOVO HSBGJóJ * f_ x i , f_ x i $ 0 –2 O y = f(x) -f_ x i , f_ x i < 0 –4 y= f_ x i = WFSJMNJõUJS x õFLMJOEF UBONMBOBO GPOLTJZPOB f GPOLTJZPOV- 2 .VUMBLEFôFSGPOLTJZP- OVO NVUMBLEFôFSGPOLTJZPOVEFOJS OVOVO Ì[FMMJLMFSJOJ LVM- G Y = EFOLMFNJOJ TBóMBZBO Y EFóFSMFSJ LSJUJL | |MBOBSBL Y2- 4 - 3 = 0 | |OPLUBMBSES Z = G Y GPOLTJZPOV LSJUJL OPLUBMB- EFOLMFNJOJOÀÌ[ÑNLÑ- SBHËSFEÐ[FOMFOFSFLQBS¿BMGPOLTJZPOBEËOÐõ- NFTJOJOFMFNBOTBZT- UÐSÐMÐS %BIB TPOSB CV QBS¿BM GPOLTJZPOB HËSF OCVMBMN GPOLTJZPOVOVOHSBGJóJ¿J[JMJS ]Y2- ] GPOLTJZPOV- ±SOFóJO y OVOHSBGJôJJMFZ= 3 4 EPôSVTV GBSLM | | G Y = Y - 1 + Y + GPOLTJZPOVOV QBS¿BM OPLUBEB LFTJöUJôJO- 3 x EFO]Y2 - 4| = 3 veya GPOLTJZPOBEËOÐõUÐSÐQHSBGJóJOJ¿J[FMJN –2 O 2 ]Y2 - 4| - 3 = 0 denk- MFNJOJO ÀÌ[ÑN LÑ- Y- 1=jY=LSJUJLOPLUBES NFTJFMFNBOMES 'POLTJZPOLJSJUJLOPLUBTOBHËSF EÐ[FOMFOJSTF x-1+x+2 ; x$1 f_ x i = * -x+1+x+2 ; x < 1 2x + 1 ; x H 1 f_ x i = * FMEFFEJMJS 3 ; x < 1 ÖRNEK 2 &MEFFEJMFOQBS¿BMGPOLTJZPOBHËSFHSBGJóJBõB- | | | | G Y = Y- + Y+ óEBLJHJCJPMVS y GPOLTJZPOVOVO HSBGJôJOJ ÀJ[FSFL BMBCJMFDFôJ FO LÑ- 3 y = f(x) ÀÑLEFôFSJCVMBMN – 1 y 'POLTJZPOVO HÌ- 2 x 5 SÑOUÑ LÑNFTJ [ ß O1 BSBMôES %PMBZTZ- –2 O 3 x MBBMBCJMFDFôJFOLÑ- ÀÑLEFôFSUJS ôJNEJEFG Y =Y-WF ÖRNEK 3 | | | | H Y = G Y = Y- GPOLTJZPOMBSOHSBGJLMF- | | | | G Y = Y+ - Y- 1 SJOJ¿J[FSFLBSBMBSOEBLJJMJõLJZJGBSLFEFMJN GPOLTJZPOVOVOBMBCJMFDFôJLBÀGBSLMUBNTBZEFôF- SJWBSES y y y = f(x) y = |f(x)| O2 x –2 2 O2 x ôFLJMEFLJHSBGJLMFSEFOGBSLFEJMFDFóJÐ[FSF y 'POLTJZPOVOVO HÌSÑOUÑ 4 Z=G Y GPOLTJZPOVOVOYFLTFOJOJOBMUOEBLBMBO LÑNFTJ [-4, 4] BSBMôES G –3 LTNMBSOOYFLTFOJOFHËSF ZBOTNBTBMOBSBL O1 JO BMBCJMFFôJ UBN TBZ EF- x ôFSMFSJUBOFEJS | |Z= G Y GPOLTJZPOVOVOHSBGJóJFMEFFEJMFCJMJS –4 32 1. 4 2. 5 3. 9
www.aydinyayinlari.com.tr '0/,4÷:0/-\"3 1. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 4 ÖRNEK 7 x-4 -3 =k 3- x+3 f_ x i = EFOLMFNJOJO GBSLM LÌLÑ PMEVôVOB HÌSF L OJO Ee- ôFSBSBMôOCVMVOV[ x2 - 4 GPOLTJZPOVOVO UBON LÑNFTJOEF LBÀ GBSLM UBN TB- y ZWBSES 3 | | | |3 - Y+ 3 $ 0 j Y+ 3 # 3 j -3 #Y+ 3 # 3 j -6 #Y# 0 y=k Y4 -âj Y- Y+ âjYâWFYâ-2 -6, -5, -4, -3, - PMNBLÑ[FSF UBNTBZWBSES O1 x 7 (SBGJôJOZ=LEPôSVTVZMBGBSLMOPLUBEBLFTJöFCJMNF- TJJÀJO< k <PMNBMES ÖRNEK 5 ÖRNEK 8 ôFLJMEFZ=G Y =Y +Y-GPOLTJZPOVOVOHSBGJóJWF- f_ x i = x - 2 - 4 SJMNJõUJS GPOLTJZPOVOVO HSBGJôJ JMF Y FLTFOJ BSBTOEB LBMBO LBQBMCÌMHFOJOBMBOLBÀCS2 EJS y y = f(x) O x y 8.4 –2 1 4 = 16 –2 O 2 #VOBHÌSF g_ x i = x· f_ x i 2 x GPOLTJZPOVOVOHSBGJôJ- 6 f_ x i OJÀJ[FMJN. y x ÖRNEK 9 2 | | | | G Y = Y+ + Y- WFH Y = 1 GPOLTJZPOMBSOOHSBGJLMFSJOJOLFTJNOPLUBMBSOOBQ- –2 O 1 TJTMFSJUPQMBNLBÀUS 1 –2 ÖRNEK 6 y 4 + k + (-2 - k) = 2 2- x+3 6 G GPOLTJZPOVOVO HSB- GPOLTJZPOVOVOFOHFOJöUBONLÑNFTJOJCVMBMN GJôJ | | | |2 - Y+ 3 $ 0 j Y+ 3 # 2 j -2 #Y+ 3 # 2 –2–k –2 O 4 4+k x 4+^-2h j -5 #Y# -1 = 1, 2 Y = EPôSVTVOB HÌSF TJNFUSJLUJS %PMBZTZMB BQTJTMFS UPQMBNEJS 4. L 6. -5 #Y# –1 33 7. 6 8. 16 9. 2
TEST - 16 .VUMBL%FôFS'POLTJZPOVWF(SBGJLMFSJ 1. G< - ] A3ZFUBONM 4. y | | G Y Y- - 2 GPOLTJZPOVO HÌSÑOUÑ LÑNFTJ BöBôEBLJMFSEFO x IBOHJTJEJS O \" <- > # <- > $ < > –4 –2 % <- ' & < ' õFLJMEFLJHSBGJLBöBôEBLJMFSEFOIBOHJTJJMFJGB- EFFEJMFCJMJS | |\" Z= Y+ | |# Z= Y- | |$ Z- =Y | |% Z= Y - | | & Z= Y + 2. f (x) = * - 2x - 1, x ≤ 1 GPOLTJZPOVWFSJMJZPS - 3x + 6, x > 1 | |Z G Y GPOLTJZPOVOVOHSBGJôJWFYekseniy- 5. f(x) = 1 MFTOSMBOBOLBQBMCÌMHFOJOBMBOLBÀCS2EJS a- x+1 \" 7 # 9 $ % 15 & 15 GPOLTJZPOVOVO FO HFOJö UBON BSBMô -4, 2 ) 4 4 42 PMEVôVOBHÌSF BLBÀUS \" # $ % & 3. y = f (x) = 1 - x2 + x 6. f (x) = x + x - 2 x +1 x x-2 GPOLTJZPOVOVOHSBGJôJBöBôEBLJMFSEFOIBOHJTJ- EJS A) y B) y GPOLTJZPOVOVOHSBGJôJBöBôEBLJMFSEFOIBOHJsi- EJS 1 x 11 x A) y B) y –1 2 2 x 2 1 O1 O O 2 O2 –2 –2 C) y D) y 1 O 1 x C) y D) y O –1 x 2 x 2 –1 O2 O2 –2 –2 E) y Ox E) y 2 x –1 1 2 –1 O –2 1. \" 2. D 3. B 34 4. \" 5. C 6. B
.VUMBL%FôFS'POLTJZPOVWF(SBGJLMFSJ TEST - 17 1. f_ x i = x - a - 4 4. 3FFMTBZMBSEBUBONM GPOLTJZPOVOVOHSBGJôJYFLTFOJOJJLJGBSLMOPL- | | | |G Y = Y- - Y+ tada LFTUJôJOFHÌSF BOOBMBCJMFDFôJLBÀGBSLM UBNTBZEFôFSJWBSES GPOLTJZPOVOVO BMBCJMFDFôJ LBÀ UBOF UBN TBZ EFôFSJWBSES \" # $ % & \" # $ % & 2. y 5. ôFLJMEFHSBGJLGGPOLTJZPOVOBBJUUJS O1 3 x y 3 3 2 –3 O3 x :VLBSEBLJ HSBGJL BöBôEBLJ GPOLTJZPOMBSE BO y = f(x) IBOHJTJOFBJUUJS ff x p = 2x2 - 4 | |\" Z=Y- Y- | |# Z=Y- Y - x | |$ Z=Y+ Y - | |% Z= Y- -Y EFOLMFNJOJ TBôMBZBO Y EFôFSMFSJOJO ÀBSQN LBÀUS | | | | & Z Y - Y- \" # - 2 3 $ 2 3 % -& 3. G3- {1} A3 f (x) = x2 - x x-1 GPOLTJZPOVOVOHSBGJôJBöBôEBLJMFSEFOIBOHJTJ- | | | |6. G3A3 G Y Y - Y- 1 + EJS GPOLTJZPOVO VOHSBGJôJBöBôEBLJMFSEFOIBOHJTJ- A) y B) y EJS –1 x 1 1 A) y B) y 3 O O1 x 3 –1 –1 O 14 O1 4 x x C) y D) y x C) y D) y –1 O 1 3 1 –1 x 3 x Ox O1 4 O –1 –1 1 E) y x E) y 1 3 O1 x O1 –1 1. C 2. \" 3. B 35 4. C 5. B 6. C
TEST - 18 .VUMBL%FôFS'POLTJZPOVWF(SBGJLMFSJ 1. G3A3 G Y = x + 2 - 5 4. | |G3Z3 G Y = - Y+ + GPOLTJZPOVOVOFOCÑZÑLEFôFSJLBÀUS PMEVôVOBHÌSF G Y = 4 denkMFNJOJOÀÌ[ÑNLÑ- \" # $ % & NFTJOFEJS \" \\- ^ # \\- - ^ $ \\- -1 } % \\- ^ & \\- - - ^ y 5. Y BSBMôOEB 2. y = f(x) G Y x - x - 6 5 3 x GPOLTJZ POVBöBôEBLJMFSEFOIBOHJTJOFFöJUUJS Oa b \" # -Y+ $ Y- G Y | Y-B| + | Y- b | % Y- & Y+ GPOLTJZPOVOVOHSBGJóJZVLBSEBWFSJMNJõUJS #VOBHÌSF a - b ifadeTJOJOEFôFSJLBÀUS a+b \" - 3 # 3 $ 5 % - 5 & - 2 6. G3={} A3 55 3 33 x-2 f (x) = x-2 -x 3. Z | Y| - GPOLTJZPOVOVOHSBGJôJBöBôEBLJMFSEFOIBOHJTJ- EJS JGBEFTJOJOHSBGJôJBöBôEBLJMFSEFOIBOHJT JEJS A) y B) y A) y B) y 3 x 3 x 2 –2 O x 1 –2 x –1 O 1 2 1 O2 O 12 C) y D) y C) y D) y O2 x O x 1 x 1 x –2 O 12 y –1 O 1 2 2 E) –2 E) y x 1 x 2 O 12 O 1. E 2. \" 3. D 36 4. C 5. D 6. \"
.VUMBL%FôFS'POLTJZPOVWF(SBGJLMFSJ TEST - 19 | |1. Z= Y +Y I I I I4. G Y Y- +Y + HSBGJôJJMFZ=BEPôSVTVOVOEÌSUGBSLMOPLUBEB GPOLTJZPOVOVOHSBGJôJBöBôEBLJMFSEFOIBOHJTJ- LFTJöNFTJOJ TBôMBZBO B UBN TBZ EFôFSMFSJOJO EJS UPQMBNLBÀUS A) y B) y 5 5 \" # $ % & x O2 x O2 2. x2 - 1 C) y D) y f (x) = x 5 5 x2 - 1 O2 O2 GPOLTJZPOVOVO HSBGJôJ BöBôEBLJMFSEFO IBOHJTJ- x x EJS A) y B) y E) y 3 1 x 1 x O2 x 5 –1 O 1 O –1 –1 1 –1 C) y D) y 1 1x O x –1 1 O –1 –1 E) y O x –1 1 5. f (x) = x2 - x2 - 4x + 4 –1 GPOLTJZPOVOVOHSBGJôJBöBôEBLJMFSEFOIBOHJTJ- EJS y A) y B) y 3. 2 2 O2 O2 2 –2 x x O1 x C) y D) y O2 õFLJMEFLJHSBGJLBöBôEBLJGPOLTJZPOMBSEBOIBO- 2 x x HJTJOFBJUPMBCJMJS 2 x I I\" Z Y- 1 +Y- 1 O I I I I# Z Y- 1 + Y + 1 I I I I$ Z Y + Y- -Y –2 I I% Z Y++ I YI I I& Z Y-+ IYI E) y O2 –2 1. \" 2. B 3. E 37 4. \" 5. B
TEST - 20 .VUMBL%FôFS'POLTJZPOVWF(SBGJLMFSJ | | | |1. G Y = Y+ + Y- 3. f (x) = *–x + 1 , x ≥ 0 GPOLTJZPOVWFSJMJZPS GPOLTJZPOV QBSÀBM GPOLTJZPO PMBSBL ZB[MSTB x , x<0 BöBôEBLJGPOLTJZPOMBSEBOIBOHJTJFMEFFEJMJS | |#VOBHÌSF H Y = G Y +GPOLTJZPOVOVO Z 2x x≥2 HSBGJôJBöBôEBLJMFSEFOIBOHJTJPMBCJMJS ] \" f_ x i = [ 4 - 2 < x < 2 ] A) y B) y \\ - 2x x # –2 –2 2 3 Z 4 x≥2 O x ] -2≤x<2 2 # f_ x i = [ 2x x O2 x < –2 ] - 4 C) y D) y \\ 3 2 Z 3x + 4 x≥2 2 ]] 1 $ f_ x i = [ x + 6 - 2 < x < 2 O1 x O1 ]] \\ - 3x + 2 x≤-2 x % f_ x i = ( 2x x ≥ 0 E) y - 2x x < 0 2 Z x + 2 x>2 ] & f_ x i = [ 4 ] -2≤ x≤ 2 O 12 x \\ - x - 2 x<-2 | |2. G Y = | Y-| - GPOLTJZPOV QBSÀBM GPOLTJZPO PMBSBL ZB[MSTB 4. G Y =Y+ | 1 - | Y| | BöBôEBLJGPOLTJZPOMBSEBOIBOHJTJFMEFFEJMJS GPOLTJZPOVO HSBGJôJ BöBôEBLJMFSEFO IBOHJTJ- EJS Z x-5 x≥5 ] ] - 2≤ x<5 \" f^ x h = [ 5 x ]] x-1 1≤ x < 2 A) y B) y \\ 1- x x<-1 1 1 Z ] x - 5 x≥5 O O f_ i ] 5 - x 2≤ x<5 – 1 1 x – 1 1 x [ 2 2 2 2 # x = ] x + 1 - 1≤ x < 2 ] - x - 1 x<-1 \\ Z ] 5 - x x≥5 C) y D) y $ f_ x i = ] x – 5 2≤x<5 1 [ x –1 O ] -x - 1 - 1≤ x < 2 ] –1 O 1 x \\ x + 1 x < –1 –1 1 –1 2 Z ] 5 - x x≥5 % f_ x i = ] x - 5 2≤x<5 E) y [ ] x + 1 - 1≤ x < 2 –1 1 ] \\ - x - 1 x<-1 O 1 Z x ] –1 x - 5 x≥5 2≤x<5 & f_ x i = [ 5 - x x<2 ] \\ x + 1 1. \" 2. B 38 3. C 4. D
.VUMBL%FôFS'POLTJZPOVWF(SBGJLMFSJ TEST - 21 1. y=x+ x–2 3. f (x) = x – x – 2 x2 - 4x + 4 GPOLTJZPOVOVOHSBGJôJBöBôEBLJMFSE FOIBOHJTJ- EJS fonkTJZPOVOVOHSBGJôJBöBôEBLJMFSEFOIBOHJsi- EJS A) y B) y A) y B) y O1 x 1 x 3 –2 O1 2 x 1 O2 x O –1 2 C) y D) y C) y D) y x 2 x O 3 –1 –1 O 1 O 12 –2 –1 O1 3 x E) x2 E) y –1 1 y O2 x 1 2 x O –1 4. G[ - ] Z3 G Y = | Y+| - | Y-| GPOLTJZPOVOVOHSBGJôJJMFYFLTFOJBSBTOEBLB- MBOCÌMHFOJOBMBOLBÀCJSJNLBSFEJS \" # $ % & | |2. Z= | Y- 1| -Y | |5. Z= | Y+ 1 | -+ | Y| GPOLTJZPOVOVOHSBGJôJBöBôEBLJMFSEFOIBOHJTJ- GPOLTJZPOVOVOHSBGJôJBöBôEBLJMFSEFOIBOHJTJ- EJS EJS A) y B) y A) y B) y 1 1 1 x –1 O 1 x 3 –1 x O1 x –1 O 1 2 2 O1 2 3 2 2 –1 C) y D) y D) y C) y 1 x 1 1 1 O 11 x3 O 1 x 1O 2 O1 x 2 2 2 2 E) y x E) y x 1 1 O1 1 O1 2 1. C 2. D 39 3. C 4. E 5. B
TEST - 22 .VUMBL%FôFS'POLTJZPOVWF(SBGJLMFSJ 1. y 4. x - 2 x - 2 = 6 2 FöJUMJôJOEF Y JO BMBCJMFDFôJ EFôFSMFSJO UPQMBN 3 x O LBÀUS –4 \" 40 # 28 $ % 38 & 33 3 õFLJMEFLJHSBGJLBöBôEBLJGPOLTJZPOMBSEBOIBO- HJTJOFBJUUJS | | | | | | | |\" Z= Y+ - Y - # Z= Y- + Y - | | | | | | | |$ Z= Y- - Y - % Z= Y+ - Y - | | | | & Z= Y- - Y 2. G Y = k - x - 4 5. Y<PMNBLÑ[FSF GPOLTJZPOVOVO FO HFOJö UBON LÑNFTJ [3, 5] | | G Y = Y- 1 + x + x - 1 BSBMôPMEVôVOBHÌSF LLBÀUS GPOLTJZPOVBöBôEBLJMFSEFOIBOHJTJOFFöJUUJS \" Y # -Y $ Y+ \" # $ % & % -Y & | | | |3. G Y = Y + Y- 1 GPOLTJZPOVBöBôEBLJMFSEFOIBOHJTJOFFöJUUJS \" f (x) = * 2x - 1, x ≥ 1 2x + 1, x < 1 Z 2x - 1, x≥1 ]] # f (x) = [ 1 , 0 ≤ x < 1 ]] \\Z - 2x + 1, x<0 | |6. G Y =- Y ]] 1 , ≥1 x GPOLTJZPOVOVOHSBGJôJJMFYFLTFOJZMFTOSMCÌM- HFOJOBMBOLBÀCS2EJS $ f (x) = [ 2x - 1, 0 ≤ x < 1 ]] \" # $ % & \\Z x -1 , x<0 ]] -1 , x>1 % f (x) = [ 2x - 1 , 0 ≤ x ≤ 1 ]] Z\\ 1 , x<0 ]] 3x - 1, x≥1 & f (x) = [ 2x - 1, 0 ≤ x < 1 ]] \\ x - 1, x<0 1. C 2. \" 3. B 40 4. B 5. B 6. \"
www.aydinyayinlari.com.tr '0/,4÷:0/-\"3 1. MODÜL ·/÷7&34÷5&:&)\";*3-*, #÷-&õ,&7&5&34'0/,4÷:0/ TANIM ÖRNEK 1 G= { - } \" # $CPõLÐNFEFOGBSLMCJSFSLÐNFPMNBL H= { - - } Ð[FSF PMEVôVOBHÌSF HPGGPOLTJZPOVOVCVMVOV[ G \" Z # H # Z $ GPOLTJZPOMBS WFSJMTJO \" LÐNFTJOJO FMFNBOMBSO G WF H GPOLTJZPOMB- gof = {(1, 1), (2, 4), (3, -2)} SZBSENZMB$LÐNFTJOJOFMFNBOMBSJMFFõMFõ- UJSFO GPOLTJZPOB G JMF H GPOLTJZPOMBSOO CJ- ÖRNEK 2 MFöLFGPOLTJZPOVEFOJSWF HPG Y õFLMJOEF HËTUFSJMJS#VSBEBGGPOLTJZPOVOVOEFóFSLÐNF- G H3Z R TJJMFHGPOLTJZPOVOVOUBONLÐNFTJFõJUUJS G Y =Y+ H Y =Y - PMEVôVOB HÌSF GPH Y WF HPG Y GPOLTJZPOMBS- HPGHCJMFõLFGEJZFPLVOVS OOLVSBMMBSOCVMVOV[ HPG\"Z$ HPG Y =H[G Y ]UJS Af B g C xyz (gof)(x) = z GPH Y =G H Y =G Y2 - 2) = Y2- 2) + 1 =Y2- 3 HPG Y =H G Y =H Y+ 1) = Y+ 1)2 - 2 =Y2 +Y- 1 #JSÌSOFLJMFBÀLMBZBMN %m/*m A = {- } #= { } ve C = {- } 'POLTJZPOMBSEB CJMFõLF JõMFNJOJO EFóJõNF LÐNFMFSJJ¿JO Ë[FMMJóJ ZPLUVS :BOJ IFSIBOHJ JLJ G WF H GPOLTJ- G\"Z# G Y =Y ve ZPOVJ¿JOGPH=HPGPMNBL[PSVOEBEFóJMEJS H#Z$ H Y =Y-GPOLTJZPOMBSJ¿JOHPG #JSGGPOLTJZPOVOVOCJSJNGPOLTJZPOJMFCJMFõLF- GPOLTJZPOVOVõFNBJMFHËTUFSJQLVSBMOCVMBMN TJZJOFGGPOLTJZPOVEVS A f B gC GPI = IPG=G –1 0 –1 01 0 ÖRNEK 3 14 3 2 G H3Z R G Y =Y+ H Y =Y- gof C PMEVôVOBHÌSF GPH L =FöJUMJôJOJTBôMBZBOLEF- A ôFSJOJCVMBMN –1 –1 f(g(k)) = 7 j f(k - 3) = 7 j 2(k - 3) + 1 = 7 j k = 6 00 13 2 HPG \"Z$ HPG Y =H G Y f_ x i = x2 4 & g_ x2 i = x2 - 1CVMVOVS g_ x i = x - 1 41 1. {(1, 1), (2, 4), (3, –2)} 2. Y2m Y2 Ym3. 6
·/÷7&34÷5&:&)\";*3-*, 1. MODÜL '0/,4÷:0/-\"3 www.aydinyayinlari.com.tr ÖRNEK 4 ÖRNEK 7 G H3Z R 5BONMPMEVôVEFôFSMFSJÀJOG Y- 2 ) =Y+ 5 ve G Y =Y+OGPOLTJZPOVWFSJMJZPS gf 3x + 4 p = x - 4 GPG Y =NY+ 2 PMEVôVOBHÌSF N+OUPQMBNOCVMBMN PMEVôVOBHÌSF HPG OFEJS GPG Y =NY+ 8 jG Y+ n) = Y+ n) + n =Y+ 4n =NY+ 8 jN= 9 , n = 2 jN+ n = 11 g ( f ( 1 ) ) = g ( 14 ) = 4 ÖRNEK 5 ÖRNEK 8 ôFLJMEFHFS¿FLTBZMBSEBUBONMGWFHGPOLTJZPOMBSOO 1 - x, x ≥ 0 HSBGJLMFSJWFSJMNJõUJS f ( x ) = * x2, x < 0 PMEVóVOBHËSF y y I. GPGPG JGBEFTJOFZFFöJUUJS y = f(x) II. ( fofof ) ( - JGBEFTJOFZFFöJUUJS 2 1 x I. f ( f ( f ( 5 ) ) ) + f ( f ( -4) ) = f ( 16 ) = -15 O y = g(x) II. f ( f ( f ( -3 ) ) ) = f ( f ( 9 ) ) = f ( -8 ) = 64 Ox –1 –2 #VOBHÌSF ÖRNEK 9 GPH - + GPH - ++ GPH + GPH G Y =Y+ g (x) = * x2 , x ≥ 2 UPQMBNOCVMBMN x+2, x < 2 ( fog )(-100) = f ( g (-100) ) = f ( 1 ) = 2 ( fog )(-99) = f ( g (-99) ) = f ( 1 ) = 2 GPOLTJZPOMBSWFSJMJZPS\"öBôEBLJMFSJCVMVOV[ h ( fog )(-1) = f ( g (-1) ) = f ( 1 ) = 2 I. HPG II. GPH ( fog )(0) = f ( g (0) ) = f ( 0 ) = 0 ( fog )(1) = f ( g (1) ) = f ( -1 ) = -2 I. g ( f ( 1 ) ) = g ( 5 ) = 25 ( fog )(100) = f ( g (-100) ) = f ( -1 ) = -2 II. f ( g ( 1) ) = f ( 3 ) = 9 5PQMBNES ÖRNEK 10 ÖRNEK 6 GEPôSVTBMGPOLTJZPOPMNBLÑ[FSF G H3Z3 GPH Y =H Y -H Y +GPOLTJZP- GPG Y =Y+ OVWFSJMJZPS PMEVôVOBHÌSF G OJOBMBCJMFDFôJEFôFSMFSJCVMBMN #VOBHÌSF G EFôFSJOJCVMBMN G Y =BY+Cj GPG Y =G G Y =B BY+C +C G H Y = [H Y ]2 - 2.[H Y ]+ 2 jG Y =Y2 -Y+ 2 j a2Y+BC+C=Y+ 6 a = 2 jC= 6 jC= 2 jG Y =Y+ 2 j f ( 2 ) = 6 f(3) = 9 - 6 + 2 = 5 a = -2 j -C= 6 jC= -6 jG Y = -Y- 6 j f ( 2 ) = -10 4. 11 5. 0 6. 5 42 7. 4 8. –15, 64 9. 25, 9 10. 6, –10
www.aydinyayinlari.com.tr '0/,4÷:0/-\"3 1. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 11 5FST'POLTJZPO 3x - 1 , x / 0 (mod 2) TANIM f (x) = * x2 + 1 , x / 1(mod 2) G\"Z#CJSFCJSWFËSUFOGPOLTJZPOVWFSJMTJO Y`\"J¿JOG Y =ZJLFOG-1 Z =YPMVZPSTBG-1 g (x) = * x + 1 , x tek say› 2x + 3 , x çift say› # LÐNFTJOEFO \" LÐNFTJOF UBONM CJS GPOLTJ- ZPOEVS#VEVSVNBõBóEBLJHJCJHËTUFSJMFCJMJS GPOLTJZPOMBSWFSJMJZPS\"öBôEBLJMFSJCVMVOV[ AB I. GPH II. HPG f III. HPH m *7 HPG xy I. f ( g ( 5 ) ) = f ( 6 ) = 17 II. g ( f ( 7 ) ) = g ( 50 ) = 103 AB III. g ( g ( -1 ) ) = g ( 0 ) = *7H G = g (11) = 12 x=f–1(y) y = f(x) ÖRNEK 12 f–1 | |G3Z3 G Y = Y- PMEVóVOBHËSF ( f o f )(k)= 5 r GJMFG-1GPOLTJZPOMBSOOHSBGJLMFSJ Z=Y B¿PSUBZ EPóSVTVOBHËSFTJNFUSJLUJS FöJUMJôJOJTBôMBZBOLEFôFSMFSJOJCVMBMN r G-1 -1 =G r GPG-1 =G-1PG= I ICJSJNGPOLTJZPO | |k - 3| - 3| = 5 j |k - 3| - 3= 5 v | |k - 3 - 3 = -5 |k - 3| = 8 |k - 3| = -2 k = -5, 11 q ÖRNEK 14 %m/*m A = {- } #= {- }LÐNFMFSJJ¿JO G\"Z# G Y =Y GPOLTJZPOVOVJODFMFZFMJN 'POLTJZPOMBSEB CJMFõLF JõMFNJOJO CJSMFõNF Ë[FMMJóJWBSES#VË[FMMJL ¦Ì[ÑN Af GP HPI = GPH PI B PMBSBLJGBEFFEJMJS –1 –1 00 11 2 ÖRNEK 13 f–1 A B G H I3Z3 –1 G Y = H Y =Y - I Y =Y-Y+ 1 –1 0 0 1 GPOLTJZPOMBSJÀJO[ GPH PI] EFôFSJOJCVMBMN 1 2 GPH PI=GP HPI j [ GPH PI](17) =G HPI = 3 ôFNBEBOHËSÐMEÐóÐÐ[FSF GGPOLTJZPOVCJSFCJSEJS GBLBU ËSUFO EFóJMEJS %PMBZTZMB G-1 JO UBON LÐ- NFTJOEFB¿LUBFMFNBOLBMBDBLUS#VOBHËSF G-1 CJSGPOLTJZPOEFóJMEJS#BõLBCJSEFZJõMFGGPOLTJZP- OVOVOUFSTJZPLUVS 11. 17, 103, 3, 12 12. –5, 11 13. 3 43
·/÷7&34÷5&:&)\";*3-*, 1. MODÜL '0/,4÷:0/-\"3 www.aydinyayinlari.com.tr ÖRNEK 15 ÖRNEK 17 A = {- } #= { }LÐNFMFSJJ¿JO \"öBôEBUBONWFEFôFSLÑNFMFSJWFSJMFOGPOLTJZPO- I\"Z# I Y =Y GPOLTJZPOVOVJODFMFZFMJN MBSEBOIBOHJTJZBEBIBOHJMFSJOJOUFSTJWBSES ¦Ì[ÑN B B h–1 A I. G/Z/ G Y =Y+ 1 h II. G;Z; G Y =Y- 0 0 –1 III. G;Z/ G Y =Y + 1 A 1 1 0 *7 G3Z3 G Y =Y - 1 7 G3+ Z R+ G Y =Y +Y+ –1 7* G;Z;+G Y =Y -Y+ 0 1 4BEFDF**CJSFCJSWFÌSUFOPMEVôVOEBOUFSTJWBSES ôFNBEBO HËSÐMEÐóÐ Ð[FSF I GPOLTJZPOV ËSUFOEJS %m/*m GBLBUCJSFCJSEFóJMEJS%PMBZTZMBI-1UBONLÐNF- TJOJO CJS FMFNBOO CJSEFO GB[MB FMFNBOMB FõMFZF- #JSGPOLTJZPOVOUFSTJOJOFõMFõUJSNFLVSBMOCVM- DFLUJS#VOBHËSF I-1CJSGPOLTJZPOEFóJMEJS#Bõ- NBLJ¿JOBõBóEBLJBENMBSJ[MFOFCJMJS LBCJSEFZJõMFIGPOLTJZPOVOVOUFSTJZPLUVS r Z=G Y LVSBMOEBYZFSJOFZ ZZFSJOFYZB- GWFIGPOLTJZPOMBSCJSMJLUFEÐõÐOÐMEÐóÐOEFGPOL- [MS TJZPOVO UFSTJOEFO CBITFEFCJMNFL J¿JO TBEFDF CJ- SFCJSMJóJOWFZBTBEFDFËSUFOMJóJOZFUFSMJHFMNFEJóJ r &MEFFEJMFOFõJUMJLUFZZBMO[CSBLMS GPOLTJZPOVOVO IFN CJSF CJS IFN EF ËSUFO PMNBT r 4POFõJUMJLUF ZZFSJOFG-1 Y ZB[MS HFSFLUJóJGBSLFEJMJS G3Z3 G Y =Y+ GPOLTJZPOVOVOUFSTJ- ÖRNEK 16 OJOFõMFõUJSNFLVSBMOCVMBMN G 3 Z 3 BöBôEB HSBGJôJ WFSJMFO GPOLTJZPOMBSEBO Z=G Y =Y+ 1 jY=Z+ 1 j y = x - 1 IBOHJMFSJOJOUFSTJWBSES 2 I. y II. y ZZFSJOFG-1 Y ZB[MSTB f–1_ x i = x - 1 PMVS 2 x xO III. y IV. y xO x ÖRNEK 18 VI. y O G3- {1} Z R - {} V. y f_ x i = 2x + 1 x-1 GPOLTJZPOVOVOUFSTJOJOFöMFöUJSNFLVSBMOCVMVOV[ Ox x O y= 2x + 1 &x= 2y + 1 & y = –1 ^ x h = x+1 x-1 y-1 f x-2 ***WF7*CJSFCJSWFÌSUFOPMEVôVOEBOUFSTJWBSES 16. *** 7* 44 17. II 18. f–1 ^ x h = x + 1 x-2
www.aydinyayinlari.com.tr '0/,4÷:0/-\"3 1. MODÜL ·/÷7&34÷5&:&)\";*3-*, %m/*m ÖRNEK 21 G3- {B} Z R - { b } f_ x i = 2x - 5 6ZHVO UBON BSBMLMBSOEB WFSJMFO BõBóEBLJ x-2 GPOLTJZPOMBSJ¿JO PMEVôVOBHÌSF (1fo44fo2. .4.4o3f)_ 3 iEFôFSJOJCVMVOV[ r G Y =BY+ b j f–1_ x i = x - b 2019 tan e a r G Y =BYj f–1_ x i = x f = f -1PMEVôVOEBOG = 1 a r f_ x i = ax + b & f–1_ x i = - dx + b cx + d cx - a r f_ x i = ax + b & f–1_ x i = cx - b ca ÖRNEK 19 ÖRNEK 22 G -Þ ] Z [ Þ G Y =Y -Y+ G3- {B} Z R - {b} f_ x i = x + 3 PMEVôVOBHÌSF f-1 Y JCVMVOV[ x+4 y = Y- 2)2 + 3 jY= (y - 2)2 + 3 jY- 3 = (y - 2)2 CJSFCJSWFÌSUFn fonksJZPOPMEVôVOBHÌSF B+CUPQ- MBNOCVMVOV[ j x-3= y-2 & x - 3 = - y + 2 & –1 ^ x h = 2 - x-3 f Y+ 4 = 0 jY= -4 = a, f-1 Y = - 4x + 3 Y- 1 = 0 x-1 ÖRNEK 23 jY= 1 =Cj a +C= -4 + 1 = -3 G3Z3 G Y = Y+ -PMNBLÐ[FSF f-1( - EFôFSJOJCVMVOV[ Y+ 1)3 - 2 = -10 j Y+ 1)3 = - 8 jY+ 1= -2 jY= - 3 ÖRNEK 20 ÖRNEK 24 GUBONMPMEVóVBSBMLUBCJSFCJSWFËSUFOCJSGPOLTJZPOPM- 5BONMPMEVôVBSBMLUB NBLÐ[FSF ff x - 1 p = x2 + 1 3 x2 + 2 f_ x i = kx + 12 WFG Y =G-1 Y 3x - 4 PMEVôVOBHÌSF ff 1 pEFôFSJOJCVMVOV[ 3 PMEVôVOBHÌSF G L EFôFSJOJCVMVOV[ kx + 12 = 4x + 12 & k = 4, f^ x h = 4x + 12 3x - 4 3x - k 3x - 4 f^ k h = f^ 4 h = 7 2 x-1 1 2 2 +1 5 = &x=2& = 33 22 + 2 6 19. –3 7 45 21. 1 22. 2 - x - 3 23. –3 5 20. 24. 6 2
·/÷7&34÷5&:&)\";*3-*, 1. MODÜL '0/,4÷:0/-\"3 www.aydinyayinlari.com.tr %m/*m ÖRNEK 28 6ZHVOLPõVMMBSEBUBONMG HWFIGPOLTJZPOMB- GWFH3Z3UBONM CJSFCJSWFËSUFOGPOLTJZPOMBSPM- SJ¿JO NBLÐ[FSF r GPH -1 =H-1 PG-1 r GPHPI -1 =I-1 PH-1 PG-1 (fog) –1 (x) = x WF G-1PI Y =Y r GPH=IjG=IPH-1WFH=G-1 PIPMVS 4 ÖRNEK 25 PMEVôVOBHÌSF h (x) EFôFSJOJCVMVOV[ G3Z3 f_ x i = 2x + 1 H Y =Y+ g–1 (x) 3 ( f-1PI Y =YjG Y =I Y PMEVôVOBHÌSF f o g-1 )-1 EFôFSJOJCVMVOV[ ^ fog h–1 ^ x h = x & ^ fog h^ x h = 4x, a x \" g–1 ^ x h k ( fog-1 )-1 = gof-1PMEVôVOEBOH G-1( 3 ) ) = g ( 4 ) = 14 4 –1 = –1 ^ x h & f^ x h = 4.g–1 ^ x h f1(g44(g2 4(4x3))) 4.g I h^ x h f^ x h –1 ^ x h & == 4.g =4 g–1 ^ x h g–1 ^ x h g–1 ^ x h ÖRNEK 26 5BONMPMEVôVEFôFSMFSJÀJO G Y =Y+WF GPH Y =Y- 1 PMEVôVOBHÌSF H Y JCVMVOV[ -1 ^ x h = x-3 & –1 og) = g PMEVôVOEBO ÖRNEK 29 4 f >f o (f ôFLJMEFZ=G Y- GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS y I g^ x h = d x - 3 no ^ 6x - 1 h = 3x - 2 42 ÖRNEK 27 2 y = f(2x – 1) x 5BONMPMEVôVEFôFSMFSJÀJO –4 –2 1 f (1 - x) = 4x - 1 WF G-1PH Y- =Y+ O 3 3 –3 PMEVô VOBHÌSF H Y JCVMVOV[ #VOBHÌSF G–1( -3 ) +G LBÀUS ( f-1PH Y-1 ) =Y+ 3 jH Y- 1 ) =G Y+ 3 ), Y= 3 j f ( 5 ) = 2 dx\" x+1 n Y= -4 j f ( -9 ) = -3 j f-1(-3) = -9 2 f ( 5 ) + f-1( -3 ) = 2 - 9 = -7 - 4x - 13 H Y =G Y+ 4 ) = , 3 GGPOLTJZPOVOEBYZ -Y- 3) 25. 14 3x - 2 - 4x - 13 46 28. 4 29. –7 26. 27. 2 3
www.aydinyayinlari.com.tr '0/,4÷:0/-\"3 1. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 30 ÖRNEK 32 ôFLJMEFGGPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS ôFLJMEFZ=G Y+ GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS y y 5 3 y = f(x+1) 4 x 3 –2 O 2 –1 1 4 1 x –1 O 1 2 3 4 #VOBHÌSF G-1 G-1 B+ =FöJUMJôJOJTBôMBZBO #VOB HÌSF GPG Y - 2 ) = EFOLMFNJOJO LÌLMFS BEFôFSJLBÀUS UPQMBNLBÀUS f-1(f-1(3a + 1)) = 4 j f-1(3a + 1) = f(4) = 0 G G Y- 2 ) ) = 3 jG Y- 2) = 0 j 3a + 1 = f(0) = 4 j a = 1 f ( 5 ) = f ( 2 ) = f(-1) =PMEVôVOEBOYEFôFSMFSJUPQMBN 741 + + =4 333 ÖRNEK 31 ÖRNEK 33 G Y CJSJODJEFSFDFEFOCJSGPOLTJZPOEVS ôFLJMEFZ=G Y GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS y h(x) y 4 3 y = f(x) g(x) –2 O 3 x –2 5 –3 x O 3 –5 f(x) õFLJMEFWFSJMFOMFSFHÌSF #VOBHÌSF (1fo44fo2. .4.4o3f)_ 5 iEFôFSJLBÀUS G-1PIPH-1 - + GPH 2019 tan e UPQMBNOOEFôFSJLBÀUS GJOJÀJOEFôFSMFSJZB[MSTB 0: ;, 3;,;-;2< , :0;, 3;,;-;2<, . . .öFLMJO- EFUFLSBSFUUJôJHÌSÑMÑS÷TUFOJMFOEFôFS-2 EJS ( f-1PIPH-1 )(-5) = f-1 I H-1 ( -5 ) ) ) = f-1 I = f-1 ( 4 ) = 0 32 (fog)(0) = f ( ( g ( 0 ) ) = f(-5) = 3 32 32 0+ = 33 30. 1 32 47 32. 4 33. –2 31. 3
TEST - 23 #JMFöLFWF5FST'POLTJZPO 1. Z=G Y TBCJUGPOLTJZPOEVS 4. G3åZå3 GPG Y å=Yå- GPG = PMEVôVOB HÌSF G BöBôEBLJMFSEFO IBOHJTJ PMEVôVOBHÌSF GPGPG EFôFSJLBÀUS PMBCJMJS \" - # - $ - % & \" # $ % & 5. R Z3ZFUBONM G1 Y =Y G Y = x G Y = x GO Y = x 2 3 n 2. f (x) = * 2x + 1 x > 1 GPOLTJZPOMBSJ¿JO 3x x ≤ 1 x 720 Z ax x>2 G1PGPPGO Y = ]] g (x) = [ 1 x+1 ]] 2 x≤2 PMEVôVOBHÌSF OLBÀUS \\ \" # $ % & GPOLTJZPOMBSWFSJMJZPS HPG å= PMEVôVOBHÌSF BLBÀUS \" - # - $ % & 6. ôFLJMEFGWFHGPOLTJZPOMBSOOHSBGJLMFSJWFSJMNJõUJS y y = f(x) y 4 y = g(x) 2 2 –1 2x –1 x O O 3. f (x) = x - 3 WF GPG N = - N GPH Y- = PMEVôVOBHÌSF YLBÀUS 2 PMEVôVOBHÌSF G N LBÀUS & \" 1 # $ 3 % & 2 2 \" - # - $ % 1. E 2. E 3. B 48 4. \" 5. D 6. \"
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