Algebra II, Chapter 7. Homework 12/5/2016. Harding Charter Prep Dr. Michael T. Lewchuk. Section 7.1 nth roots and Rational Exponents

Transcription

1 Algebr II, Chpter 7 Hrdig Chrter Prep Dr. Michel T. Lewchuk Test scores re vilble olie. I will ot discuss the test. st retke opportuit Sturd Dec. If ou hve ot tke the test, it is our resposibilit to mke it up i withi the llowed time ccordig to school rules. It is probbl i YOUR best iterest to tke the test before Thksgivig. Homework Ch 6 homework will o loger be ccepted Ch 7 homework will receive utomtic poit deductio per d if tured i lte without vlid reso There will be two questio mii quiz o ech dte tht homework is due. Grde will be 0, or 0 out of 0. The questios m come directl from the homework Sectio 7. th roots d Rtiol Epoets Properties of Epoets Summr 0 b 0 m m m 0 m m m m m Product Rule Quotiet Rule Zero Epoet Negtive-Epoet Power to Power Rule Simplifig Epoetil Epressios. Remove ll pretheses usig power/multiplictio rules. Remove powers to powers usig the power to power rule. Combie like bses so tht ech bse ppers ol oce. Rewrite ll bses to the epoet zero s. Rewrite equtio so tht ll epoets re positive b b b b Power of Product Power of Quotiet The full simplified swer should hve o pretheses, ol positive epoets d ech bse should pper ol oce

2 The Pricipl th root of Rel Number b mes tht b 6, mes tht 6 If is eve the is 0 d b 0 If is odd the d b re rel umbers Fidig the th roots of Perfect th powers If is odd the If is eve the Emples Simplif: ( ) or The Defiitio of Also,, 0 is iteger d It is rel umber if The Defiitio of m m = m m It is rel umber if is rtiol umber d m Also,, 0 m m m The Reltioship betwee th Roots d Rtiol Epoets

3 Emples Rtiolizig Deomitors 7 6 Rtiolize the deomitor: Emples Rtiolize the deomitor: The Product d Quotiet Rules for th Roots Sectio 7. Properties of Rtiol Epoets b b b, b 0 b

4 Emple Simplif: Emple Simplif: Emple Simplif: Emple Simplif: 8 0 Notice tht the ide reduces o this lst problem. Simplif the epoetil epressio () (b) (c) (d) Simplif the epoetil epressio 0 () (b) (c) (d)

5 Simplif: () (b) (c) (d) Simplif: () (b) (c) (d) Sectio 7. Power Fuctios d Fuctio Opertios h( ) 0 Domi :, g ( ) Domi restrictios: These two vlues must be ecluded Domi:,,, Emple Fid the domi of the fuctio f( ) 0 D :,, Emple Fid the domi of the fuctio f ( ) 0 D :,

6 Domis of Some Fuctios f ( ) f ( ) f ( ) Domi:, Domi :, or Domi: 0, f( ) Domi:,0 0, Chpter 6 test retkes Sturd School, this Sturd Mod 7:0 m Tuesd :0 pm 7 f( ) 9 7 Domi:,,, Mii Quizzes Doors Choose Everoe i the room is required to ttempt the d s mii quiz. If ou hve bee bset d thik it should ot cout, idicte so er our me d I will mke the fil decisio. Filure to ttempt the questios will be tke ito ccout i m decisio. Aoe writig Thk You letter will receive 0 poit bous o the 7. homework ssigmet. Mteril icluded Pecil Shrpeer Cordless Mouse Equtio editor, llowig color chges The Algebr of Fuctios The Sum, Differece, Product d Quotiet of fuctios Give f ( ) d g( ) the ew fuctio h( ) c be formed b combig the first two fuctios usig the bsic lgebric opertors.. Sum f ( ) g( ) ( f g)( ) h( ). Differece f ( ) g( ) ( f g)( ) h( ). Product ( fg)( ) ( f g)( ) h( ) f f ( ). Quotiet ( ) h( ), but g( ) 0 g g( ) The domi of h( ) is the itersectio f ( ) d g( ). D D D h f g Determiig Domis Whe Addig or Subtrctig Fuctios The domi of f g is the set of ll rel umbers tht re commo to the domi of f d the domi of g. Thus we must fid the domis of f d g before fidig their itersectio. Suppose f ( ) d g( ) the ( f g)( ) Now for their domis. f ( ) ; 0 d g( ) ; 0 So the domi for the sum of the fuctios is which i itervl ottio is, Cotiued o et slide 6

7 Cotiutio of the sme problem. The grph of ( f g)( ) cofirms tht the domi of this fuctio is, Determiig Domis whe Multiplig Fuctios The domi of f g is the set of ll rel umbers tht re commo to the domi of f d the domi of g. Thus we must fid the domis of f d g before fidig their itersectio. Suppose f ( ) d g( ) the ( f g) ( ) Now for their domis. f( ) g ( ) 0 0 So the domi for the product of the fuctiosis 0, which i itervl ottio is,0 0,, Determiig Domis whe Dividig Fuctios If the fuctio f g the domi before simplifig. c be simplified, determie Emple; f ( ) d g( ) f g Emple If f ( ) g( ) Fid ech of the followig: ( f g)( ) ( f g)( ) 7 7 ( fg)( ) 0 7 f but i ; Therefore D:,, g f ( ) g ;D:,, Emple If f ( ) g( ) 9 Fid the domi of the followig: ( fg)( ) f ( ) g Emple If f ( ) g( ) Fid the domi of the followig: ( fg)( ) f ( ) g 7

8 Emple If f ( ) g( ) 6 Fid the domi for: Composite Fuctios ( f g)( ) ( ) f g f g We red this equtio s " f of g of is equl to " We cll f ( g( )) the compositio of the fuctio f with g, or composite fuctio. This composite fuctio c be writte s f g. The domi of f g is,0 0,, Emple Give f( ) d g ( ). Fid f g ( ) g ( ) b. Fid the domi of f g( ) The domi of f g( ) is ssess g ( ) 0,0 0,, ssess f g( ) 0; ; ; Emple Give f ( ) d g( ). f g. Fid f g = ( ) b. Fid the domi of f g ssess g ( ) 0 ssess f g( ) 0 9 The domi of f 0,9 9, g( ) is 8

9 Decomposig Fuctios Not covered i the tetbook Not required h( ) c be writte s the compositio of wht two fuctios? g()= g( ) f ( ) h ( ) f g Emple Epress h ( ) s compositio of two fuctios: h( ) 6 Emple Epress h ( ) s compositio of two fuctios: h ( ) 9 6 Simplif: 88 () (b) (c) (d) Divide 8 () (b) (c) (d) 6 9

10 Fid the domi of the fuctio f( ) 67 (),,7 7, (b),,7 7, (c), 7, (d),,7 7, If f ( ) d g( ), Fid ( f g)( ) () (b) ( ) (c) (d) Fid the domi of f () (b) (c) (d) g if f ( ) d g( ) 0,,,,0 Let f() = 7. Q + 8, g() =, h() = + f( ) h ( ) , D :, 7. Q 6 f( ) 8 f ( ) 8 Let f() = + 8, g() =, h() = + From Q 8 roots () ( ) ( ) ()() D :,,,, 7. Q 8 f ( ) ( ) f 8 g ; f ; g g (8) 8 () () 8 D :, 0

11 Sectio 7. Iverse Fuctios How to Fid Iverse Fuctio Fidig the Iverse of Fuctio Fid the iverse fuctio of f ( ). f ( ) Replce f ( ) with :, Iterchge d : R ( Solve for : eplce with f ) : f ( ) Emple Fid the iverse of f 7 7, replce f ( ) with 7, swp d 7, solve for 7 7 ( ), replce with f f ( ) 7 Emple Fid the iverse of f ( ) f ( ), replce f ( ) with, swp d, solve for f f ( ), replce with ( )

12 Emple Fid the iverse of f( ) ( ) or ( ) ( ) ( ) f The Horizotl Lie Test Ad Oe-to-Oe Fuctios Horizotl Lie Test b d c re ot oe-to-oe fuctios becuse the do t pss the horizotl lie test. Emple Grph the followig fuctio d tell whether it hs iverse fuctio or ot. f ( ) Frid s pl Fiish 7. otes, if ecessr Review 7. b 7. b quiz Review to 7. review pcket will NOT be grded.

13 Pret Fuctio of Squre Root f ( ) Grphs of f d f - is the iverse of with domi restrictios Pret Fuctio of Cube Root f ( ) is the iverse of There is reltioship betwee the grph of oe-to-oe fuctio, f, d its iverse f -. Becuse iverse fuctios hve ordered pirs with the coordites iterchged, if the poit (,b) is o the grph of f the the poit (b,) is o the grph of f -. The poits (,b) d (b,) re smmetric with respect to the lie =. Thus grph of f - is reflectio of the grph of f bout the lie =. Emple If this fuctio hs iverse fuctio, f( ) the grph it s iverse o the sme grph. Emple If this fuctio hs iverse fuctio, the grph it s iverse o the sme grph. f ( )

14 Emple If this fuctio hs iverse fuctio, the grph it s iverse o the sme grph. f ( ) Uique Propert of f d f - f f ( ) This c be used to prove tht two fuctios re iverses of ech other Applictios of Iverse Fuctios The fuctio give b f()=(9/)+ coverts degrees Celsius to equivlet temperture i degrees Fhreheit.. Is f oe-to-oe fuctio? Wh or wh ot? F=f()=(9/)+ is to becuse it is lier fuctio. b. Fid formul for f - d iterpret wht it clcultes. 9 f ( ) The Celsius formul coverts degrees Fhreheit ito Celsius. Replce the f() with Swp d the solve for C f ( ) ( ) Fid equtio for f ( ) give tht f ( ) () - f ( ) (b) - f ( ) (c) - f ( ) (d) - f ( ) Fid equtio for f ( ) give f ( ) ( ) - () f ( ) Sectio 7. - (b) f ( ) (c) (d) - f ( ) - f ( ) - Grphig Squre Root d Cube Root Fuctios

15 Pret Fuctio of Lie D :, :, f ( ) R f ( ) Pret Fuctio of Absolute Vlue D :, R : 0, Pret Fuctio of Qudrtic f ( ) D :, R : 0, Pret Fuctio of Cubic D :, f ( ) R :, Pret Fuctio of Squre Root f ( ) D : 0, R : 0, Pret Fuctio of Cube Root f ( ) D :, R :, is the iverse of with domi restrictios is the iverse of

16 Trsformtios of Squre Root D :,0 f ( ) : 0, Trsformtios of Squre Root R f ( ) D : 0, R :,0 Trsformtios of Squre Root f ( ) Trsformtios of Squre Root D : 0, R :, f ( ) 6 D : 6, R : 0, Trsformtios of Squre Root f ( ) D :, R :, Trsformtios of Squre Root f ( ) D : 0, R : 0, 6

17 Trsformtios of Squre Root f ( ) D :, R : 0, Trsformtios of Cube Root D :, R :, f ( ) Trsformtios of Cube Root D :, f ( ) R :, Trsformtios of Cube Root D :, f ( ) R :, Trsformtios of Cube Root D :, f ( ) R :, Trsformtios of Cube Root D :, f ( ) R :, 7

18 + + Sectio 7.6 Solvig Rdicl Equtios Solvig rdicl equtios Steps for solvig rdicl equtios:. Isolte the rdicl or epoet o oe or both sides of the equtio.. Use the iverse fuctio to elimite the rdicl or epoet.. Solve the equtio.. Check our swer STEP STEP STEP STEP Wh do we hve to check solutios wheever risig equtio to eve power? Some solutios re clled etreous or flse solutios becuse risig both sides of equtio to the sme power might possibl itroduce issue due to the ± we kow bout eve roots. Therefore, it is criticl tht ou check our solutios i mthemtics whe doig this tpe of problem. Emple If we squre both sides, we obti 6 6 or This ew equtio hs two solutios, - d. B cotrst, ol is solutio of the origil equtio, =. For this reso, whe risig both sides of equtio to eve power, check proposed solutios i the origil equtio. Etr solutios m be itroduced whe ou rise both sides of rdicl equtio to eve power. Such solutios, which re ot solutios of the give equtio re clled etreous solutios or etreous roots. Emple Give Solve for () If we check this solutio however: Therefore, it is criticl tht ou check our solutios i mthemtics whe doig this tpe of problem. 8

19 Emple Solve 0 d check our swers: 0 0 ( )( ) Two posssible solutios Check soluios i origil equtio or 0 Emple Solve: ( 6) ( 8 0) 0 ( 0)( ) 0 Two posssible solutios 0 or Trsformtios of Squre Root Trsformtios of Squre Root Trsformtios of Squre Root Trsformtios of Squre Root 9

20 Trsformtios of Squre Root Trsformtios of Squre Root Trsformtios of Squre Root Trsformtios of Cube Root Trsformtios of Cube Root Trsformtios of Cube Root 0

21 Trsformtios of Cube Root Trsformtios of Cube Root

22

23