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# College Algebra MATH 1310

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This 26 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 1310 at University of Houston taught by Staff in Fall. Since its upload, it has received 9 views. For similar materials see /class/208413/math-1310-university-of-houston in Mathmatics at University of Houston.

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Date Created: 09/19/15

Slopeintercth form of a line y mx b is the slope is the yintercept Example 19 Put the equation 5 y 2x 15 in slopeintercept form Then nd the slope and y intercept Slope y intercept Section 12 7 L8 7 Page 1 of 5 To graph a line from the equation using the slope and y intercept do the following steps 1 2 Put the equation in slopeintercept form Find the y intercept and plot Use the slope to nd a second point Connect the dots Section 12 7 L8 7 Page 2 of 5 Example 20 Sketch the graph of the line With equation 5 y 2x 15 Step 1 Put the equation in slopeintercept form Solve for y Step 2 Find the y intercept Step 3 Find a second point on the graph using the slope Step 4 Connect the dots Section 12 7 L8 7 Page 3 of 5 Example 21 Sketch the graph of the line With equation 3x 4y 2 4 Step 1 Put the equation in slopeintercept form Solve for y Step 2 Find the y intercept Step 3 Find a second point on the graph using the slope Step 4 Connect the dots Section 12 7 L8 7 Page 4 of 5 Example 22 Sketch the graph of the line With equation 3y 2 2x Section 12 7L8 7 Page 5 of 5 Mathl 3 10 Chapter 2 Section 25 Additional Equations Solve by Factoring To solve an equation by factoring set one side of the equation equal to 0 if necessary and apply the ZeroProduct Property Example 1 Find all solutions real or complex x3 4x2 4x l6 0 Example 2 Find all solutions real or complex x3 5x2 6x O Example 3 Find all solutions real or complex x5 49x3 Solve an Equation with the Variable Under a Radical Sign To solve an equation in which the variable appears under a radical sign isolate the radical on one side of the equation and raise both sides to a power that will eliminate the radical Note When solving equations by raising both sides to an even power you must check the proposed solutions Some proposed solutions may be extraneous Example 4 Find all solutions real or complex V3x10 x Example 5 Find all solutions real or complex x5 xx 2 1 Example 6 Find all solutions real or complex Nx2x 2 Solve by Substitution In solving an equation quadratic in form a technique of substitution can be used In this case an appropriate substitution will transform the given equation into a quadratic equation Previous techniques for solving quadratic equations can then be used Example 7 Use substitution to find all solutions real or complex x4 4x2 3220 Example 8 Use substitution to find all solutions real or complex 5xllx20 Example 3 Find three consecutive even integers Whose sum is 222 Let x the rst even integer the second even integer the third even integer The integers are and Section 22 7 Page 3 of 12 Seeliun 31 Dr ni nnufz runen39un Page l at IE e Arunetlnn fls arule that asslgns to eaeh elementx ln a setA ealledthe dnmzin off exactly one element ealled x ln a set The elements ln sets A andB are real mbers 5 s mbol x ls read quotfunquot and ls the value offatx The range offls the set of posslble values of x as x vanes throughout the domaln A To evaluateat anumber substltute the number forx m the de nltlon off x r4 The dnmzin ofls the set R ofreal numbers slnee x4 20 for all real numbers x the range offls the set0w To evaluate fat the numbers 73 0 and 5 substltute those numbers forx lnr r4 A 3 81 Thevalueof ate3ls81 00 0 Thevalueoffat ls 5 5 625 The value off at 5 rs 625 Example 1 Evaluate the functionx 2x2 3x74 at the numbers 2 and Snlutinn tn Example 1 72 272237274 24 74 87674 Seclinn 31 Dr ni nnufz Func nn P2g22uf m xii jdti fwt 4 lt2gt2lt2gt23lt2gt74 4 10 JE2J gmw 223J574 43J 74 35 Example 2 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 Smee l lt 2 l 1710 Smee222 23 256511 ExampleA State whether ornot eael ofthe mapplngs shown below represents afunctlon 2 0 1 2 t 5 5 3lt 339 7 A B 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 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 12Lhen 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 Dummy Func nn Pigesun Example 5 Forthe function gwen byfx 5x72 evaluate72 0 f8 1 am Snlntinn m Example 5 To nd72 substitute 72 for x x 5x7 2 A72 592 2 7107 2 712 To nd0 substitute 0 forx To nd3 substitute 3 forx x x 2 lt3gt5lt372 15 13 To nd L substitute i forx 171 171 Seclinn 31 Dr ni nn uf Func nn Pige ul l fx5x72 1 1 Haw ail Example 7 Forthe function gwen byfx x71 73 evaluate71 4 anda Snlntinn m Example 7 To nd71 subsmute 71 for x x x712 3 fH 02 73 722 73 47 1 To nd4 substitute 4 forx ltxgtltx71gt2 3 104471 73 3 973 6 Seclinn 31 Dr ni nnufz Func nn P2ge7uf To nd substitute for x x x712 3 4391 To nd a substitute a forx ltxgtltx71gt273 041170273 27214473 72172 Examples 3 2 f 0 Forthefuncnon gwen by x X K lt evaluate74f72 6x1 xfoU 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 Seclinn 31 Dr ni nnufz Func nn ragesuns Dumzin ura Funninn Smee 2 lt0 tofmd fez subsutute 2 forxm the expresssxon 73x2 72 7372 44 712 IfoU then the ya1ue off at x15 gwen by 6x1 Since 0 20 to fmd my subsutute 0 forx 1n the expresssxon 6x1 0 60 1 1 1 Since 6 20 to fmd 6 subsutute s forx 1n the expresssxon 1 6661 61 Domain of a Function Ifthe domam ofa functlon has notbeen exphmtly stated then by eonyentmnthe domam The funcuonf gwen by x Lquot 15 not de ned when the denommator 15 0 The X denommator x e 1 15 0 when x 1 fxs not de ned at x 1 The domam off 15 x x1 Inmtervalnotauonthe domam offxs gxyenbyew1 The funcuonfngen by fx qu 15 not de ned for those values ofx where x e 1 15 negatwe 51 t square mot of a negatw n mber 15 not a real number Thus the domam off 15 x1xe1zox1x21 In mterval notauon the domam off 15 gwen by W Seelinn 3 nvmalnnra runen39nn Pageynns Example 1 Flnd the domam ofthe funeuonx Snlutinn tn Example 1 The funetlon ls not de ned whenthe denomlnator ls 0 en x The denomlnator x2 5x6ls 0 when x3x2 0 Thusls not de ned wh or x 7 domam of 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 t domam offls xllsex 20xlmexxzznpoxlezzsxszz The d lnterval notatlon ls 744 omam ln Example 3 Flnd the domam othe funeuonx 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 denomlnator ls 0 The denomlnator lx2 ls 0 whenx 72 Thus the domam offlsx xgt72 l Seclinn 31 Danniinf Func nn Page lnunu Example 4 Flndthe domam othe functionx s xe 2 Express the answer m lnterval holauoh Snllltinn m Example For x 2 e to be a eal number Exe 2 cannot be negatlve Solve the lnequallty Exe 22 0 font to ndthe domam 8x420 Exe22202 8x22 Ex 2 Vi x2 b m The domam m lnterval holauoh 15 300 Example 5 Flndthe domam othe functionx 373K Express the answer m lnterval notatl Snllltinn m Example For r473x to be areal number 473x cannot be negatlve Solve the lnequallty 4 7 3x 2 0 for x to mat the domam o Seclinn 31 Dumzinuf Func nn Fig 11 quE 473x20 473x742074 3x243 The domam m interval hexamh Ts 00 Example 5 Fmdthe domam othe fuhehOhx 1 Express the answer 47 9x m interval hexamh Snllltinn m Example 5 Fmdthe values ofx where the denominator 0 by solving the equamh 4 7 9x 0 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 Dumzinuf Func nn Page lzunu Example Frndthe domam of the functlonfx Express the answer 78x7 m rnterval notatron Snlntinn tn Example 7 Frndthe values ofx where the denomrnatorrs 0 by sellmg the equatron x2 7 x2 a Ex 7 0 xrlxxrmw 1 0r x77 0 xrl eol xr77e 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 Snlntinn tn Example For x 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 9 74 3 List the mtervels determined by 74 and 3 3024 743 300 G so 74 3 Check the stghs ofthe factors x4 and x73 on the mterval 7094 by usmg atestvalue 75 is m the mtehval 70074 For x e 5471lt0 and x73 57378 lt0 ngh ofx 4 7 tgh of stgh of x4xe3 Seclinn 31 Dumzinufz Func nn Pig am Show the results on the number hhe o o Cheekthe srghs ofthe factors r4 and73 on the mterval 743 by usmg atest value 015m the rhterval 743 Forr0 r4 044 gt0 and x7307373lt0 Srghofr4 home 7 Srgh ofx4x73 7 Show the results on the number hhe Check the srghs ofthe factors x4 and73 on the mterval 1w by usmg atest value 4 rs m the rhterval 300 X4 48gt0 and x734731gt0 Srgh ofx 4 rgh ofx Srgh of x4x73

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