Nme: Algebr II Honors Pre-Chpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble to think bckwrds Evlute the Rdicls below! n= (perfect squres) n= (perfect cubes) n= n= n= 7 8 9 0 7 8 9 0 0 0 0 9 8 9 8 7 79 000 8
Section : Evlute nth Roots nd Use Rtionl Eponents The nth root of n n for ny integer n greter thn n is the of the rdicl is clled the nd cn be be rel number Rtionl Eponents Let m n be n nth root of, nd let m be positive integer n n m n ( ) ( ) m n, 0 m n m n ( ) ( ) m Emple : Evlute epressions with rtionl eponents b 8 Emple : Approimte with clcultor b c ( ) Emple : Solve Equtions using nth roots n is odd n is even b c d e 8 f ( 8) 00 g ( ) h 8 0
Let n represent positive integer Under wht conditions will n eqution of the form n hve no rel solutions? Under wht conditions will n eqution of the form n hve ectly one rel solution? Under wht conditions will n eqution of the form n hve two rel solutions? Section : Apply Properties of Rtionl Eponents Recll the properties of eponents:
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8 Section : Solve Rdicl Equtions Emple Emple Emple Emple Emple ( 7)
9 Equtions with Etrneous Solutions Etrneous solutions re solutions tht emerge from the process of solving n eqution, but re not vlid solution to the contet of the originl problem Emple Emple 8 0 b b
70 Section : Grph Squre Root nd Cube Root Functions Grph: y Grph: f ( ) Domin: Rnge: Domin: Rnge: Trnsformtions tke functions nd trnsform them through verticl/horizontl shifts nd reflections An esy wy to understnd trnsformtions is to think of functions s hving two prts; n inside nd n outside Most of the time, inside refers to the prt of the eqution inside prentheses or under symbols Inside trnsformtions move functions horizontlly while outside trnsformtions move functions verticlly Trnsformtions inside re lwys the opposite from wht is given to you nd trnsformtions outside re tken s they re given Let s look t n emple of generl trnsformtion f ( ) h k or f ( ) h k, etc
7 Here is wht ech prt does: Let s look t the outside first coefficient * if is between 0 nd, it is considered verticl compression * if is lrger thn, it is considered verticl stretch * if is negtive, it is reflection over the -is k constnt: gives you verticl shift up/down *if it is+k, it will move up tht mny *if it is k, it will move down tht mny Now let s look t the inside Remember tht inside trnsformtions re the opposite of wht ppers h constnt: gives you horizontl shift left/right *if it is + h, it will move left tht mny *if it is - h, it will move right tht mny Grph Rdicl Functions: Emple : Grph ) Stte the nme of the function: b) Stte the trnsformtion: c) Grph: Originl Function Stretch/Compression Vert/Hor /Reflection Shifts d) Domin: Rnge: (Intervl Nottion)
Emple : Grph 8 ) Stte the nme of the function: b) Stte the trnsformtion: c) Grph: Originl Function Stretch/Compression Vert/Hor /Reflection Shifts 7 d) Domin: Rnge: (Intervl Nottion) Emple : Grph ) Stte the nme of the function: b) Stte the trnsformtion: c) Grph: Originl Function Stretch/Compression Vert/Hor /Reflection Shifts d) Domin: Rnge: (Intervl Nottion)
7 BIG IDEA ABOUT DOMAIN!!! Assume when you re given function tht the domin will be ll rel numbers or, unless you see one of the following conditions: If ny of these conditions eist in the function, the domin hs to be restricted Vribles in the denomintor of frction: Set denomintor 0 nd solve Vribles underneth even rdicls or bse rised to rtionl eponent with n even denomintor: Set underneth or bse 0 nd solve If there re vribles underneth even rdicls or bse rised to rtionl eponent with n even denomintor tht re in the denomintor of frction: Set underneth or bse 0 nd solve since the denomintor cnnot equl zero! Find the domin nd write your nswer using intervl nottion f ( ) g ( ) 7 f ( ) h ( ) g ( ) h ( )
7 Section : Perform Function Opertions nd Composition Nottion: The functions f + g, f g, nd fg re defined for ech for which both f nd g re defined Tht is to sy, the domin of rithmetic combintions is the intersection of the domins of f nd g (domin of f + g) = (domin of f g) = (domin of fg) = Domin of f Domin of g The function f g is defined for ech for which both f nd g re defined nd g() 0 Tht is,(domin of f ) = Domin of f Domin of g, s long s g() 0 g Perform the following function opertions on the following functions Stte the domin of ech using intervl nottion ) f ( ) nd g( )
7 b) f ( ) nd g ( ) Domin of f: Domin of g: c) f ( ) nd g( ) Domin of intersection: Domin of f: Domin of g: Domin of intersection: Emple : Add nd subtrct functions Let f ( ) nd g( ) Find the following f ( ) g( ) b f ( ) g( ) c the domins of f g nd f g Emple : Multiply nd divide functions Let f ( ) 8 nd f ( ) g( ) g( ) b Find the following f( ) g ( ) c the domins of f g nd f g
7 Emples of nottion for composition of functions Rules for Composition of functions: Whtever is lst function in line is the first mchine You cn think of it s working from the inside, out E f(g()) would set up like this: g(f()) would set up like this:
b) c) d) d) 77 Emple : Let f ( ) nd g Wht is the vlue of f( g( ))? ( ) Finding the domin of composite function consists of two steps: Step Find the domin of the "inside" (input) function (First Mchine) Grph the domin of the inside function Step Construct the composite function Find the domin of this new function (Finl Output) Grph the domin of the composite function Use the most restrictive domin (or the intersection of the domins) The composite my result in domin not necessrily equl to the domins of the originl functions Emple : Given the following functions, find the function composition nd stte the domin in intervl nottion f ( ) g ( ) h( ) p( ) j ( ) ) f ( g( )) b) ( g f )( ) Domin of g: Domin of f: Domin of f ( g( )) : Domin of ( g f )( ) : Domin of Intersection Domin of intersection: c) f f d) g( j( ))
78 Decomposition of function Sometimes we cn write given function s the composition of two or more other functions This is clled decomposing the function It is importnt to be ble to decompose functions in lter work in clculus For emple, tke the function h() = + For this function h, find functions f nd g such tht h() = g(f()) In this cse, we could write f() = + nd g() = The rule cn chnge depending on the question being sked, for emple, if the directions sy tht h() = f(g()) nd then g() = + nd f() = For the given function h, find functions f nd g such tht h() = f(g()) Emple : h() = ( + ) ( + ) + f() = g() = For the given function h, find functions f nd g such tht h() = g(f()) Emple : h() = ( + ) f() = g() =
Prctice: Given the following functions, find the function opertion or composition nd stte the domin in intervl nottion 79 f ( ) g ( ) h( ) p( ) j ( ) e) h( ) p( ) f) h( ) g( ) g) ( p g)( ) h) h( ) p( ) i) g( p( )) j) j p k) j ( ) g ( )
80 Section : Use Inverse Functions One-to-one functions: A function where neither the -vlue nor the y-vlue repet The test for one-to-one function is the horizontl line test Like the verticl line test used to determine if reltion is function, you drw horizontl line through the grph nd if it crosses once, it is one-to-one If it crosses more thn once, it is not one-to-one
Inverse Functions: In order for function to hve n inverse tht is lso function, the originl function must be one-to-one The inverse of one-one function is obtined by switching the role of nd y, then re-solve the eqution for y The new function will be identified by the nottion f () Properties of inverse functions: 8 The domin of f = the rnge of f The rnge of f = the domin of f The inverse of f is f, tht is (f ) = f Inverse functions re symmetric bout the line y = Emple : Find the inverse of f ( ) Grph of f Domin of f: Rnge of f: One-to-one? Y N Inverse is: Grph of inverse Domin of inverse: Rnge of inverse: Is our inverse n inverse function? Y N If not, wht cn we do to mke our inverse n inverse function? Write the inverse function using the correct nottion
8 Emple : Find the inverse of f ( ) Grph of f Domin of f: Rnge of f: One-to-one? Y N Inverse is: Grph of inverse Domin of inverse: Rnge of inverse: Is our inverse n inverse function? Y N If not, wht cn we do to mke our inverse n inverse function? Write the inverse function using the correct nottion
8 Emple : Find the inverse of f ( ), 0 Grph of f Domin of f: Rnge of f: One-to-one? Y N Inverse is: Grph of inverse Domin of inverse: Rnge of inverse: Is our inverse n inverse function? Y N If not, wht cn we do to mke our inverse n inverse function? Write the inverse function using the correct nottion Emple : Verify tht f() = + nd g() = + re inverses using (f f )() & (f f)()
8 Emple : Verify tht f() = + using (f f )() & (f f)() nd g() = re inverses You Try