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# Class Note for MATH 1310 at UH 2

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This 18 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 16 views.

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Date Created: 02/06/15

Seclinn 31 Dr ni nn uf Func nn xii jdti fwt lt2gt2lt2gt23lt2gt74 24674 8674 10 JE2J gmw 223J574 43J 74 35 Pig 2 III In Seclinn u Dr ni nnufz Func nn Pageant IE Example 3 Evaluate the followlng funeuon at 721 and 2 2 x 71 lf xlt2 x f0 3x5 lf x22 Snllltinn tn Example 3 Ifx lt 2 than the value offatx ls x2 71 Ex 2 2 thenthe value offatx ls 3x 5 Smee e2lt2 72722714713 Slncellt2f1 1710 Smee222 23 256511 Example 4 State whether ornot eael ofthe mapplngs shown below represents afunctlon Seelinn u Dr ni nnufz Func nn Page IE Snlutinn tn Example 4 Desenbe the rst dlagram l ls assrgnedto 0 2 ls assrgnedto 5 3 ls assrgnedto 5 The rst mapplng represents afuncuon srnee eaen element ofA 123 ls assrgnedto exaetly one element ofB 05 Desenbe the second dlagram l ls assrgnedto 2 3 ls assrgnedto botln 5 and7 The secondmapplng does not represent a functlon srnee not every element of A 13 ls assrgnedto exaetly one element ofB 257 3m Als asslgned to two elements 5 and 7 m B Example Express the rule multrply by 5 subtraet l2 tlnen square m funeuon notatron For example subtraet 3 tlnen take tlne square root would be wntten as x Jxa3 Snlutinn tn Example 5 For tlne rst step begln wrtln x and multrply by 5 5x Next subtraet 12 from 5x 5x712 For tlne last step square 5x712 5x7122 Now use funeuon notatron fx 5x7122 Seclinn 31 Dr ni nn uf Func nn Example 5 Forms funcuon gwen byfx 5x72 evaluate72 0 3 anda Snlntinn m Example 5 To nd72 substitute 72 for x x 5x7 2 fez 592 2 71072 42 To nd0 substitute 0 forx To nd3 substitute 3 forx x x72 3 53 J2 1572 13 To nd substitute i forx a7 ail Pagesufl Seclinn 31 Dr ni nnufz Func nn Pige ul l fx5x72 a1715a17172 572a2 Example 7 Forthe function gwen byfx x71 73 evaluate71 4 anda Snlntinn m Example 7 To nd71 subsmute 71 for x x H2 3 fH 02 73 472 473 1 To nd4 subsmute 4 forx ltxgtltx71gt273 104470273 3173 973 6 Seclinn 31 Dr ni nn uf Func nn To nd substitute for x x x712 3 4 e71 To nda subsmute a forx ltxgtltx71gt273 041170273 27214473 724172 Example 8 For the function gwen by x f 0 v and 6 Snlntinn m Example 8 1h lt0 than the value of at xxs gwen by 73x2 Smce 74 lt 010 fmd A74 substitute 74 forx m the expression 73x2 74 734 7306 438 0 evaluate4 2v P2ge7uf Seetimt 31 Dentinant Func nn ragesuns Dumzin uf Funninn stnee 72 lt0 tofmd fez substatttte 72 forxm the expression 73x2 72 7372 734 712 IfoU then the ya1tte off at xls gtyen by 6x1 stnee 0 20 to ftnd my sttbshtttte 0 forx tn the expressslon 6x1 0so1 01 1 stnee s 20 to ftnd 6 sttbshtttte s forx tn the expressslon 6x1 6661 361 37 Domain of a Function Ifthe domatn ofa funetton has notbeen exphettly stated then by eonyenttonthe domatn ts the set ofall real numbers for whteh the expresston ts deftned as a real number The funetton gtyen by 331 ts not de ned when the denominator ts 0 The X denominator x e 1 ts 0 when x 1 Thusts not de ned at x The domatn off ts x1 x 1 1n tntetyat notattonthe domatn offxs gtyenbyew1g1w The funettongtyen by fx sXe1 ts not de ned at those yahtes ofx Where x e 1 ts negattye stnee the square toot of a negattye number ts not a teat number Thus the domatn off ts x1 x7120 x x21 In tntetyat notatton the domatn off ts gtyen by W Seeliun 3 nvmalnvra Func nn Pageyvns Example 1 Flnd the domam ofthe funeuonx Snlutinn tn Example 1 The functlon ls not de ned whenthe denommator ls 0 The denommator x2 5xsls 0 when x3x2 0 Thusls not de ned when x 73 or x 72 The domam off ls xl x 71x 72 The domam ln lnterval notatlons ls w3U32U2UJ Example 2 Flnd the domaln ofthe funeuonx J16 x2 Snlutinn tn Example 2 The funetlonls not de ned for those values of x where 16 ls negatlve Thus the domam offls xllsex 20xlmexxzznpoxlezzsxszz The domam ln mterval notatlon ls 744 Example 3 Flnd the domam othe funeuonx Jx2 Snlutinn tn Example 3 Begm by notlng that lx2 ls a real number for those values ofx where x 2 20 The set of values for whleh x 2 20 ls xl x272 Next note that the funetlonls not de ned when the denommator ls 0 The denommator lx2 ls 0 whenx 72 Thus the domam of ls xl x gt 72 In lnterval notatlon the domam ls 7200 Seclinn 3 Dumzinufz Func nn Page lnunu Example 4 Fmdthe domam othe functionx x 2 Express the answer m lnterval hotatror Snlntinn tn Example For rExe 2 to be areal humber 8x7 2 earmot be hegatrve Solve the lnequallty 8x7 22 0 font to hdthe domam off 8x420 Exe22202 8x22 Ex 2 Vi x2 b be The domam m mterval notatloh 15 300 Example 5 Fmdthe domam othe functionx a473x Express the answer m lnterval hotatror Snlntinn tn Example For r473x to be areal humber 473x earmot be hegatrve Solve the mequahty 4 7 3x 2 0 for x to mat the domam off Seclinn 31 Dumzinuf Func nn Fig 11 quE 473x20 473x742074 4 The domam m mcerval notation is we Example 5 Fmdthe domam othe functionx Express the answer m interval notation thmnh m Example 5 Fmdthe values ofx where the denominator 0 by solving the equauon 4 7 9x 0 479x0 The fuheuoh is de ned for each real humbem except those values ofx Where the denominators 0 Thus the domam is x x 2 The domam m mcerval notation is Seclinn 3 nnmalnnra new Page lznnu Example Frndthe domam of the funetronx X Express the answer 7 8x 7 m lnterval notatlon Snlntlnn tn Example 7 Frndthe values ofx where the denomrnatorrs 0 by solvlng the equatron x2 a 8x 7 0 x278x7 0 xrlxr7o xrlo or xrl eol The funetlon 15 de ned for each real numberx except those values ofx where the denomrnatorrs 0 Thus the domam ls x lx LX 7 The domam m rnterval notatron ls 001U17U7w Examples Frndthe domam of the funetronx Ix x712 Express the answer m rnterval notatron Snlntlnn tn Example For xi 2 to be areal number x2 x712 eannot be negatwe Solve the lnequallty x1 x7122010 nd the domam off Seclinn 31 Dumzinuf Func nn Pig ISquE x2x71220 x4xe320 Solve the conespondmg equahon x4xe3 0 MA 0 or x73 0 x474 074 xe3303 F3 Place 74 and 3 on ahumbehhhe The closed melee mdmate that the numbers sausz the mequahty x4x7 3 2 0 a o 74 3 Lxst the mtervals determmed by 74 and 3 food 743 zoo o s 74 3 Check the stghs ofthe factors x4 and x73 on the mteml 414 by usmg atestvalue 75 is m the mtehval 70074 For x 75 x4 54 71 lt 0 and x73 57378 lt0 stgh ofx4 e stgh ofxe3 e stgh of x4xe3 Seclinn 31 Dumzinuf Func nn Pig am Show the results on the number hhe 9 0 Cheekthe stghs ofthe factors x4 and73 on the mteml 743 by usmg atest value 0 is m the mterval 743 Forx0v x4044gt0 and x7307373 lt0 stgh ofx4 stgh ofx73 e stgh of x4x73 7 Show the results on the number hhe Check the stghs ofthe factors x4 and73 on the mteml 300 by usmg atest value 4 is m the mterval 300 Forx KM 48gt0 and x734731gt0 stgh ofx4 stgh ofx73 stgh of x4x73 Seclinn 31 Dumzinuf Func nn Pig 1511113 Show the results on the number lme o o 4 3 The domam othe function 5 m x 0073 0 Wm the domam m mm notation from the number lme analysts The domam m interval notation is wr4u3w Exercise Set 31 Basic Ideas For each of the examples below determine whether the mapping makes sense within the context of the given situation and then state whether or not the mapping represents a function 1 Erik conducts a science experiment and maps the temperature outside his kitchen Window at various times during the morning 57 gt 62 65 Time Temp 39F Dr Kim counts the number of people in attendance at various times during his lecture this afternoon 85 87 Time of People State whether or not each of the following mappings represents a function 3 S A B 4 2 A B 5 A B 6 0 8 39 4 A B Express each of the following rules in function notation For example Subtract 3 then square would 7 8 9 be written as fx x 32 Divide by 7 then add 4 Multiply by 2 then square Take the square root then subtract 6 10 Add 4 square then subtract 2 Complete the table for each of the following functions 11 fxx375 12 fx goo x7 4f 1 fx Find the domain of each of the following functions Then express your answer in interval notation 13 14 15 16 7 5 fx7x73 if 7 fooi x8 fxquot quot xl hxx3 x74 Exercise Set 31 Basic Ideas 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 x74 x279 gov 3xl x24 x x26x5 foo 2 fx x2 72x7 24 fx 77 2x f0 J7 hm aE goo 5 ha 47 foo J3 goo J foo 8im foo 24W m x4 J F06 Gx mHTs gx3 x2 x76 7 277t78 HIT 7 2x79 fx 5l4xi7 x 711x28 37 38 39 40 goo M hm ft m go m Evaluate the following 41 42 43 44 45 46 47 If fx5x74 find f3 Ilia f 1 fa3 fa3 faf3 If fx3xl find f6 res le NH ma form If gx x2 73x 4 find gm gH gem g9 gen 34 If ht t2 2175 find ha he hc 6 7 hm h2x 2m 2 find 3 If x x m7 f0 14 f0 mks x2 x4 If x 7x find f2 f5 Ilia f3P fP32 2x75 if x24 If fx find 37x2 if xlt 4 mi N es f0 f4 re Exercise Set 31 Basic Ideas 48 49 50 2 If foo x 4x 1f xlt72 nd 772x if x272 res f6 M N fH A7 3x2 if xlt 0 If fx 4 if 0gxlt2find x5 if x22 f0 res A2 fa f4 re 74x77 if xgil If fx x26 if 1ltxlt3find 77 if x23 f0 M fH f6 ms 14 For each of the following problems 21 Find fa b Find fah c Find Assume that no 51 52 53 54 55 56 57 58 fx7x74 fx573x fxx23 fx2x271 fxx25x72 fxx273x8 fx8 fx6 Seeiimi 31 nrrmiu39unvra Func nn Page 1 ur IE De nition of a Function Arunetinn is arule that assigns to eaeh elementx in a setA ealledthe dnmzin off eiraetlv one element ealled x in a set The elements in sets A and B are real numbers The symbol x is read quotlubequot and is the value offatx The range offls the set of allposslble values of x as x varies throughout the domain A To evaluate fat anumber substitute the number forx in the definition off The funetiongiven by fx r4 assigns to eaeh real number x the fourth power ofx The dnmain ofis the set R ofreal numbers sinee x4 20 for all real numbers x the range offls the set0w To evaluate fat the numbers 73 0 and 5 substitute those numbers forx in fx r4 73 3 81 Thevalue off ate3is81 00 0 Thevalueoffat ls 5 5 625 The value off at 5 is 625 Example 1 Evaluate the funetionr 2x2 3x74 at the numbers 2 and Snlutinn in Example 1 e2 7 2722 37274 24 67 4 87674

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