n m CHAPTER 3 RATIONAL EXPONENTS AND RADICAL FUNCTIONS 3-1 Evaluate n th Roots and Use Rational Exponents Real nth Roots of a n th Root of a
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1 CHAPTER RATIONAL EXPONENTS AND RADICAL FUNCTIONS Big IDEAS: 1) Usig ratioal expoets ) Performig fuctio operatios ad fidig iverse fuctios ) Graphig radical fuctios ad solvig radical equatios Sectio: Essetial Questio -1 Evaluate th Roots ad Use Ratioal Expoets What is the relatioship betwee th roots ad ratioal expoets? Key Vocabulary If b a, the b is the th root of a th Root of a Examples: 9 is the root of is the root of Ratioal Expoet A expoet writte i ratioal/fractioal form represets a 1/ 1 / Examples: 4 4, Aatomy of a Ratioal Expoet: a m Aatomy of a Radical: m a a 0 Let be a iteger Real th Roots of a 1 The idex,, is a iteger Example: square root, fourth root, etc. Example: No th roots: a imagiary 1 i ad let a be real umber. The idex,, is a iteger Example: cube root, fifth root, etc. real th roots: Example: 8 a a 1/ a 0 real th root: 0 0 Example: 0 0 real th root: 0 0 Example: 0 0 a 0 real th roots: Example: 4 a a 1/ real th roots: Example: 8 a a 1/ Studet Notes Hoors Algebra II Chapter Ratioal Expoets ad Radical Fuctios Page #1
2 Ex 1) Fid the idicated real th root(s) of a. a., a b. 6, a 1 Ex ) Evaluate. / a. 1 b. 4/ 8 c. 4 / d. 1 7/8 Ex ) Evaluate the expressio usig a calculator. Roud the result to the earest hudredth. a. 1/4 b. /6 c Ex 4) Solve the equatio. Roud the result to the earest hudredth, if ecessary. a. 6x 84 b. ( x 8) 10 0 Ex ) A exercise ball is made from 784 square cetimeters of material. Fid the diameter of the ball. (Use the formula S 4 r for the surface area of a sphere.) Studet Notes Hoors Algebra II Chapter Ratioal Expoets ad Radical Fuctios Page #
3 Sectio: Essetial Questio - Apply Properties of Ratioal Expoets How are the properties of ratioal expoets related to the properties of iteger expoets? Key Vocabulary Simplest Radical Form Like Radicals Simplest Ratioal Expoet Form Radicad has o perfect power factors Deomiator is 4 4 Same AND same ( ad ) Note: You must have like radicals i order to add or subtract radicals! No or expoets! (Ratioal expoets are i the deomiator.) PROPERTIES OF EXPONENTS Let a ad b be real umbers ad let m ad be ratioal umbers. a m ab a a m m m a, a 0 a a m a b, a 0 m, b 0 PROPERTIES OF RADICALS Product Property of Radicals a b Quotiet Property of Radicals a b, b 0 Radicals with Variable Expressios Remember! Whe workig with real umbers, umbered roots oly work for umbers. Sice variables ca represet positive OR egative umbers, we must cosider two cases: The idex,, is a eve iteger Example: square root, fourth root, etc. The idex,, is a odd iteger Example: cube root, fifth root, etc. x x Example: 4 4 ad Example: ad 7 Studet Notes Hoors Algebra II Chapter Ratioal Expoets ad Radical Fuctios Page #
4 Simplify the expressio. Assume all variables are positive. 1/8 /6 1/ 1/ / /4 1/ s s 80s z 11. 1/ m x y z 1. 6 r s ab 7 /4 /6 1. a c 7 p q a b 8a b 4 / 4 / Closure: Explai how to ratioalize the deomiator for a cube root. How is it differet tha ratioalizig the deomiator for a square root? Studet Notes Hoors Algebra II Chapter Ratioal Expoets ad Radical Fuctios Page #4
5 Sectio: Essetial Questio - Perform Fuctio Operatios ad Compositio What operatios ca be performed o a pair of FUNctios to obtai a third FUNctio? Key Vocabulary Power FUNctio A fuctio of the form where a is a real umber ad b is a 1/ ratioal umber. Examples: y 4 x, f ( x) 8 x A o fuctios where h( x) ( f g)( x) f ( g( x)). Compositio of FUNctios A combiatio of two fuctios where oe fuctio is performed, the the result is used to perform the secod fuctio. The domai of h is the set of all x-values such that x is i the domai of g AND gx ( ) is i the domai of f. Operatios o FUNctios Let f ad g be ay two fuctios. A h ca be defied by performig ay of the four basic operatios o f ad g. Operatio Defiitio Example: f ( x) x, g( x) x Additio h( x) f ( x) g( x) Subtractio h( x) f ( x) g( x) Multiplicatio h( x) f ( x) gx ( ) Divisio Cautio: hx ( ) f( x) gx ( ) Domai of the ew fuctio h cosists of the x-values i the domais of ad. The quotiet does ot iclude x-values for which. Ex 1) Let f ( x) 1/ x ad g( x) 11 fid the domai of hx ( ). 1/ x. Use the followig operatios to fid ( ) a. h( x) f ( x) g( x) b. h( x) f ( x) g( x) hx, the Studet Notes Hoors Algebra II Chapter Ratioal Expoets ad Radical Fuctios Page #
6 Ex ) Let f ( x) 8x ad the domai of hx ( ). g( x) /6 x. Use the followig operatios to fid ( ) hx, the fid a. h( x) f ( x) g( x) b. hx ( ) f( x) gx ( ) Ex ) A small compay sells computer priters over the iteret. The compay s total mothly reveue (R) ad costs (C) are modeled by the fuctios R( x) 10x ad C( x) 00 7x, where x is the umber of priters sold. a. Fid R( x) C( x) b. Explai the meaig of this differece. Ex 4) Let f ( x) x 4 ad g( x) x 1. a. Fid f( g( ))? b. Fid g( f( ))? Studet Notes Hoors Algebra II Chapter Ratioal Expoets ad Radical Fuctios Page #6
7 Ex ) Let f ( x) 6 domais. x ad g( x) 4x. Fid the followig compositios, the fid their a. f ( g( x )) b. g( f ( x )) c. ggx ( ( )) Ex 6) Your startig wage for your part-time job was $6 a hour. All employees get a % raise after 6 moths. You are give a additioal raise of 7-cets per hour as a reward for your outstadig work. Fid your ew hourly wage if the % raise is applied before the 7-cet raise. Fid your ew hourly wage if the 7-cet raise is applied before the % raise. Closure: Whe performig a compositio of fuctios, is the order of compositio importat? If so, which fuctio must be performed first? What is the domai of a fuctio ad how do you fid it? Studet Notes Hoors Algebra II Chapter Ratioal Expoets ad Radical Fuctios Page #7
8 Sectio: Essetial Questio -4 Use Iverse FUNctios How do you fid the iverse relatio of a give fuctio? Key Vocabulary Iverse Relatio A relatio that the iput ( ) ad output ( ). Its graph is a reflectio across the diagoal lie. A iverse relatio that has the properties of a fuctio. The iverse relatio passes the. Iverse FUNctio Fuctios f ad g are iverses of each other provided: AND Fuctio Notatio: read f iverse of x (Cautio!! This is NOT the same otatio as the expoet meaig to take a reciprocal! f x ) f ( x) 1 1 Horizotal Lie Test The iverse of a fuctio f is also a fuctio if ad oly if o itersects the graph of f oce. Iverse is Iverse is Ex 1) Fid a equatio for the iverse of the relatio y 4x. Studet Notes Hoors Algebra II Chapter Ratioal Expoets ad Radical Fuctios Page #8
9 Ex ) Verify that f( x) 4x ad 1 1 f 1 ( x) x are iverse fuctios. 4 Ex ) A small compay produces greetig cards. The cost C (i dollars) of producig greetig cards per moth ca be modeled by the fuctio C Fid the iverse of the model. Use the iverse fuctio to fid the umber of greetig cards produced i a moth i which the compay s total cost to produce the cards was $61. Ex 4) Fid the iverse of f ( x) x x, 0. The graph 1 f ad f. Studet Notes Hoors Algebra II Chapter Ratioal Expoets ad Radical Fuctios Page #9
10 Ex ) Cosider the fuctio f x ( ) x fuctio. The fid the iverse.. Determie whether the iverse of f is a Ex 6) The populatio of a tow ca be modeled by years sice P 16,00t 0.1, where t is the umber of Fid the iverse model that gives the umber of years as a fuctio of the populatio. Use the iverse model to predict the year i which the populatio of the tow will reach,000. Closure: What is the differece betwee a fuctio ad a relatio? Does every fuctio have a iverse? Does every fuctio have a iverse fuctio? Explai how to fid a iverse fuctio. Studet Notes Hoors Algebra II Chapter Ratioal Expoets ad Radical Fuctios Page #10
11 Sectio: - Graph Square Root ad Cube Root Fuctios Essetial Questio What do the graphs of square root ad cube root fuctios look like? Key Vocabulary Radical FUNctio A fuctio of the form Examples: y x, y x The most fuctio of a family of fuctios. Used for graphig with trasformatio of fuctios. Square Root Paret FUNctio Cube Root Paret FUNctio Paret FUNctio Ex 1) Graph y x, ad state the domai ad rage. Compare the graph with the graph of y x. Ex ) Graph y x, ad state the domai ad rage. Compare the graph with the graph of y x. Studet Notes Hoors Algebra II Chapter Ratioal Expoets ad Radical Fuctios Page #11
12 Ex ) Biologists have foud that the shoulder height h (i cetimeters) of a male Africa elephat ca be modeled by h 6. t 7.8, where t is the elephat s age (i years). a. Use a graphig calculator to graph the model. b. Estimate the age of a elephat whose shoulder height is 00 cetimeters. Ex 4) Graph y x 4. The state the domai ad rage. Ex ) Graph y x. The state the domai ad rage. Studet Notes Hoors Algebra II Chapter Ratioal Expoets ad Radical Fuctios Page #1
13 Sectio: Essetial Questio -6 Solve Radical Equatios Why is it ecessary to check every apparet solutio of a radical equatio i the origial equatio? Key Vocabulary Example: Extraeous Solutio solutio(s) created by raisig both sides of a equatio to the same power that must be. Procedure Solvig Radical Equatios Step 1: the radical o oe side of the equatio. Example: x 4 x Step : Raise each side of the equatio to the same power to the radical. Step : the polyomial equatio. Step 4: the solutio to elimiate extraeous solutios. Note: You may eed to repeat steps 1 ad if there is more tha oe radical! x x 8 x 11? 11 4? 8 4? Solve the equatio. Ex 1) x 9 11 Studet Notes Hoors Algebra II Chapter Ratioal Expoets ad Radical Fuctios Page #1
14 Ex ) If a 8-foot ladder is leaed agaist a wall, the height of the ladder alog the wall is give by h( x) 64 x, where x is the distace alog the floor from the base of the ladder to the wall. Estimate the distace that the ladder should be placed from the wall i order for the ladder to reach 7 feet off the floor. Solve the equatio. Ex ) / 7 6 x Ex 4) x Ex ) x 6 x 8 Ex 6) x 6 11 x Studet Notes Hoors Algebra II Chapter Ratioal Expoets ad Radical Fuctios Page #14
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