MAC1105-College Algebra

Transcription

1 . Inverse Funcions I. Inroducion MAC5-College Algera Chaper -Inverse, Eponenial & Logarihmic Funcions To refresh our undersnading of "Composiion of a funcion" or "Composie funcion" which refers o he comining of funcions in a manner where he oupu from one funcion ecomes he inpu for he ne funcion. Recall he noaion for composiion is: ( f g)( ) f [ g( )] is read " f composed wih g of " or " f of g of " or fog and ( g f )( ) g[ f ( )] is read "g composed wih f of " or "g of f of " or gof. and also recall: In general, ( f g)( ) ( g f )( ) or f[ g( )] g[ f( )] For eample: if f ( ) 5 g( ) hen, ( f g)( ) f [ g( )] f ( ) 5 ( g f )( ) g[ f ( )] g( 5) ( 5) 5 or, ( f g)( ) f [ g( )] f ( ) f ( ) 5 ( g f )( ) g( 5) g( ) ( ) You can see ha algeraicall or numericall/mahemaicall, in general, ( f g)( ) ( g f )( ) II. Onl a special class of funcions would have ( f g)( ) ( g f )( ) ;his is he Inverse Funcion. A. Definiion: Recall, in general, ( f g)( ) ( g f )( ) or f [ g( )] g[ f ( )] Onl in a ver special class of funcions, such ha he composie funcions, ( f g)( ) ( g f )( ) or f [ g( )] g[ f ( )], his is he Inverse Funcion. f ( ) dancing around waving he magic wand. B. Noaion: f ( ), he magic wand, "un-doing", equivalen o he "Undo" in programming...or equilaven o having he magic wand ha can undo he momen when he vase

2 S.Nunamaker was roken ino pieces., undo-ing he pieces of vase ack o is original form. How do I mean ha? Firsl, le's ake a look a few eamples of how o generae an Inverse Funcion: C. Generaing an Inverse Funcion Eample: f ( ) 5 o generae an inverse funcion of he original funcion, 5 wrie he funcion in his form, replace f ( ) 5 swap he and 5 hen solve for 5 his is he inverse funcion of he original f ( ), f ( ) 5 Noe ha in he noaion for he inverse funcion, f ( ), he - is no an eponen. D. If we were given wo funcions and are asked o deermine if one of he wo funcions is he inverse funcion of he oher, he rue Acid Tes for Inverse Funcion, ( f f )( ),( f f )( ) If given f ( ) 5, g( ) 5 Deermine: f [ g( )] and g[ f ( )] Therefore, If f[ g( )] g[ f( )] hen, g( ) f ( ) f ( ) g ( ) or, ( f f )( ) and ( f f )( ) E. Relaionship eween Domain and Range of a Funcion and Is Inverse Eample: Given: f ( ) 6 Find: D & R of f ( ) Find: f ( ) Find: D & R of f ( )

3 Domain of he funcion is equal o he Range of is inverse funcion whereas he Range of he funcion is equal o he Domain of is inverse funcion. F. Graphical Relaionship eween a Funcion and Is Inverse : (Swapping eween D and R roles of f ( ) and f ( ) ) Eample: f ( ) g( ) A funcion and is inverse funcion are reflecion of one anoher across he line. Horizonal line es of he original funcion as a wa o deermine if is inverse is a funcion. Now, le's generae he inverse funcion of anoher funcion: Eample: f ( ) graph of f ( ) and g( ) f ( ) In his inverse funcion, one value would generae wo values and is herefore no a funcion, or is no a funcion of. f ( ) has an inverse relaionship wih ha of f ( ), u i is no a funcion relaionship ecause f ( ) fails he verical line es. Graphicall, recall ha a funcion and is inverse are reflecion aou he line, so he verical line es of one is equivalen o he horizonal line es of he oher. Therefore, if he original funcion passes he horizonal line es (for each value, here is onl one value corresponded o i), hen is inverse is a funcion. Bu if he original funcion fails he horizonal line es, hen is inverse is no a funcion. This is one o one idea. The idea ha for each value

4 of i is corresponded o one value of. We can see ha when a funcion and is inverse are oh of funcional relaionship, here is he one o one idea. Tha usuall occurs when he funcion is coninuousl increasing or decreasing. G. The graphing calculaor (TI-8). Using graphing calculaor (TI-8)o quickl deermine if he inverse of our original funcion graph is a funcion using he horizonal line es.. Using TI-8, and he fac ha f [ f ( )] f [ f ( )], so graphing, ( ), if and are inverse of each oher, he composion of ( ) would ield he ideni funcion,. This would verif if one is he inverse of he oher.. Using TI-8's ale feaure o verif he swapping of and values eween funcions and heir inverses. *Place he original funcion ino. Press nd-draw-8 o ge o DrawInv. The command will appear on he home screen waiing for a parameer. Ener he locaion of he funcion o e invesigaed,.(using VARS-Y-VARS--ENTER-ENTER) o ge DrawInv Y and he graph of he funcion and is inverse will appear.. Horizonal line es of he original funcion as a wa o deermine if is inverse is a funcion. Eample: f ( ) graph of f ( ) and f ( ) f ( ) is no a funcion relaionship (failed he verical line es) f ( ) failed he horizonal line es (is inverse is no a funcion). f ( ) and f ( ) are reflecion of one anoher across he line. f [ f ( )] f [ f ( )], so graphing, ( ), if and are inverse of each oher, he composiion of ( ) would ield he ideni funcion,. This would verif if one is he inverse of he oher. Eample: f ( ), f ( ). Swapping eween D and R roles of f ( ) and f ( ) Eample: f ( ) = f ( ) H. Imporan Facs aou Inverses. If f is one-o-one, hen f eiss. The domain of f is equal o he range of f, and he range of f is equal o he domain of f.. If he poin (a, ) lies on he graph of f, hen (, a) lies on he graph of he f, so he graph of f and he graph of f are reflecions of each oher across he line.. To find he equaion for f, replace f( ) wih, inerchange and, and solve for. This gives f ( ).

5 . Eponenial Funcions A. Eponenial Funcions The funcion f ( ) a, where is a real numer, a, and a, is called he eponenial funcion, ase a. Requiring he ase o e posiive would help o avoid he comple numers ha would occur aking even roos of negaive numers. (E., ( ), which is no a real numer.) The resricion a is made o eclude he consan funcion f ( ). Eample: f ( ) 5, f ( ) ( ), 6 f ( ) (. 75) *The variale in an eponenial funcion is in he eponen. B. Graphing an eponenial funcion. Ploing poins. Graphing calculaor * Tr f ( ) ( ) *Noice he change if ase is> or < *Noice if is > or < C. Applicaion i Eample: Compound Ineres Formula, A P( ) n r for n compoundings per ear: or A P( ) n for coninuous compounding: A Pe r. a oal of $, is invesed a an annual rae of 9%. Find he alance afer 5 ears if i is compounded quarerl: r n. ( ) A P( ), ( 9 5 ) 8, 76. n. a oal of $, is invesed a an annual rae of 9%. Find he alance afer 5 ears if i is compounded coninuousl: n n 5

6 r. 9( 5) A Pe, e 8, coninuous compounding ields ineres: =9.6 D. The Numer e n A( n) ( ), as he n ges larger and larger, he funcion value ges closer o e. Is n decimal represenaion does no erminae or repea; i is irraional. In 7, Leonard Euler named his numer e. You can use he e ke on a graphing calculaor o find values of he eponenial funcion f ( ) e. Eample: Find e, e., e, e 5. 8, e 6

7 .-. Logarihmic Funcions A. Logarihm Graph: Logarihmic, or logarihm, funcions are inverse of eponenial funcions and have man applicaions. Eample: Conver each o a logarihmic equaion: 6 >log 6. log. (he logarihm, ase, of 6 ) (he power o which we raise o ge 6) e 7 log 7 e Eample: Find each of he following logarihms: log, log. log 8 log 9 log 6 log 8 8 * log a and log a a, for an logarihmic ase a B. Common Logarihm and Naural Logarihm. Common logarihms are of ase-. The areviaion log, wih no ase wrien, is used o represen he common logarihms, or ase logarihms. Eample: log 9 means log 9 log = log 7

8 . Naural Logarihms Logarihms, ase e, are called naural logarihms. The naural logarihm's areviaion is ln. Eample: ln means log e C. Changing Logarihmic Bases Change-of-Base Formula: log M log log M lnm ln Eample: log log log 66. ln log ln 869. D. Graphs of Logarihmic Funcions Graph: f ( ) log Graph: g( ) ln 5-8

9 . Evaluaing Logarihms and he Change-of-Base Theorem. Eample: 8. Eample: 5 5. ( ) Eample: ( ) 96. log log log The Produc Rule Eample: log 6 log log log ( ) log log The Quoien Rule 6 Eample: log ( ) log 6 log 8 8 n 6. log ( ) n log The Power Rule Eample: log ( ) log 6 7. log Eample: log 8 8. log Eample: log 8 8 n 9. log ( ) n Eample: log ( ). f ( ) log ln, is he naural logarihmic funcion. ln e e. ln ln e e ln. ln( uv) lnu ln v 5. log a a log 6. a a 9

10 7. Changing Logarihmic Bases Change-of-Base Formula: log M log log M lnm ln Eample: log log log 66. ln log E. Tr hese: ln 869. Epress each as a sum, difference, or muliple of logarihms. log 5. log 5. log 9. log 6 5. log log log 5( ) 5 8. log 7 9. log ( ) 8. log 5. ln 5 7. log 75 Epress each as he logarihm of a single quani. log a log c. log 9 log 5 5. log e log n. e log a log 5 5. ln ( ) 6. ln ln ln z

11 .5. Eponenial and Logarihmic Equaions A. To solve an eponenial equaion, firs isolae he eponenial epression, hen ake he logarihm of oh sides and solve for he variale. Eample: Solving Solving e 7 lne ln 7 ln Solving e 5 6 e 55 ln e ln55 ln 55. Solving e e ( e )( e ) e, e if e lne ln ln. 69 if e lne ln B. To solve a logarihmic equaion, rewrie he equaion in eponenial form (eponeniaing) and solve for he variale. ln Eample: Solving ln e e e 789. Solving 5 ln ln ln ln e e. 67 ln e Solving ln ln e e e. 6 Solving ln ln( ) ln e e e e e e e ( e) e e e

12 .6 Applicaions and Models of Eponenial Growh and Deca Eample: You have deposied $5 in an accoun ha pas 6.75% ineres, compounded coninuousl. a. How long will i ake our mone o doule?. How long will i ake our mone o riple? r a. for coninuous compounding, A Pe 5e. 675, or P( ) P e k. =5e 675 To find he ime required o doule, A 5e. 675, or P ( ) e e. 675 ln ln e ln. 675 ln To find he ime required o riple, A 5 5e or P( ) 5 5e e. 675 ln ln e ln. 675 ln Douling Time, T ln, Tripling Time, T k ln k Eponenial Growh Rae, k ln T or k ln T Eponenial and Logarihmic Models A. The five mos common pes of mahemaical models involving eponenial funcions and logarihmic funcions: k. Eponenial growh: ae, or P( ) P e, k * or k is he eponenial growh rae. eample: The ale show he growh of amospheric caron dioide over ime: a. Find an eponenial model using he daa for and 75. Le he ear correspond o. Use he model o esimae when fuure levels of caron dioide will riple from he 95 level of 8 ppm.

13 Year Caron Dioide (ppm) a. Since ear corresponds o, he ear corresponds o 75., using e k e k ( ) 9 k ( ) 9 75 e ln( ) ln e k ln( ) 75k k k k he equaion of he model is: e or 75e. 6. To riple from he level of 8 ppm: ( 8) 8. hen solve for, given 8 75e e ln( ) lne ln( ) eample: How long will i ake for he mone in an accoun ha is compounded coninuousl a 5.75% o doule? Using coninuousl compounding formula: A Pe r P Pe e ln ln e I will ake aou ears o doule. eample: The projeced world populaion (in illions of people) ears afer, is. given he funcion f ( ) 679. e 6 a. Wha will e he world populaion e a he end of he ear 5?. In wha ear will he world populaion reach 8 illion?. 6( 5) a. f ( ) 679. e 7. illions e ln( ) ln e , a aou

14 k. Eponenial deca: ae, or P( ) P e, k * or k is he eponenial deca rae (E. radioacive susance) Half-life: half of he radioacive susance will cease o e radioacive wihin ha period of ime. eample: If 8 grams of a radioacive susance are presen iniiall, and.5 ears laer onl grams remain, how much of he susance will e presen afer ears? a. Using e k, o solve for k firs:. Afer k is oained, solve for a eample: Suppose ha he skeleon of a woman who lived in he Classical Greek period was discovered in 5. Caron esing a ha ime deermined ha he skeleon conained / of he caron of a living woman of he same size. Esimae he ear in which eh Greek woman.6 died. Hin: The amoun of radiocaron presen afer ears is given e, ( ),. 6 6 e. e, solve for. ( his value represens he amoun of ime efore he ear 5).. Newon's Law of cooling sas ha he rae a which a od cools is proporional o he difference C in emperaure eween he od and he environmn around i. The emperaure of he od a ime in appropriae unis afer eing inroduced ino an environmen having consan emperaure T is f ( ) T Ce k where C and k are consans. Newon's Law of cooling also applies o warming. eample: Jane ook a leg of lam ou of her refrigeraor, which is se a F, and placed i in her oven, which she had preheaed o 5 F. Afer hour, her mea hermomeer regisered 7 F. a. Wrie an equaion o model he daa. Find he emperaure 9 minues afer he leg of lam was placed in he oven. c. Jane wans o serve he leg of lam rare, whichrequires an inernal emperauer of 5 F. Wha is he oal amoun of ime i will ake o cook eh leg of lam?

15 To solve: a. when, f ( ), T 5 k ( ), Solve for C: f ( ) 5 Ce 5 C, f ( ) 7 C 5 6 ( ) Solve for k: f ( ) 5 6e 7 5 6e 8 6e k k k 8 k 8 k 8 e ln( ) lne ln( ) k. k. k The equaion is f ( ) 5 6e. Since is in unis of hours, 9 minues is =.5 hours., f (. ) e. ( ) 865. F c. If jane wans o serve he leg of lam rare, which requires an inernal emperaure of 5 F, he oal amoun of ime ha will ake o cook he leg of lam 5 5 6e 5 6e 5. e 6 5. ln( ) lne The mea emperaure will e 5 F afer aou.576 hours or aou hours 5 minues.. Gaussian model: ae ( ) c This pe of model is used in proaili and saisics o represen populaions ha are normall disriued. The sandard normal disriuion akes he form Gaussian model is ell-shaped curve. e The graph of a 5

16 5. Logisics growh model or logisic funcion: a e ( c) d ( = pop. size, = ime) a or P( ) his funcion increases oward a limiing value as. Thus, he e k horizonal asmpoe of a or, P( ) approaches a as. So, a is he limiing value in his model of limied growh. Some populaion iniiall have rapid growh, followed a declining rae of growh. This pe of growh paern is of logisics curve; i is also called a sigmoidal curve. Eample: aceria culure or spread of an epidemic Tr: In a own where populaion is 5, a disease creaes an epidemic. The numer of people N infeced das afer he disease has egun is given he funcion: 5 N ( ). 99. e 6 a. Graph he funcion. How man are iniiall infeced wih he disease ( )? c. Find he numer infeced afer das, 5das, 8 das? d. Using his model, can ou sa wheher all 5 people will ever e infeced? 6. Logarihmic model: ln( a ), log ( a ) E.: On he Richer scale, he magniude of R of an earhquake of inensi I: R B. Take a look a he asic shapes of hese graphs:. e. e. e I log I. e 5. ln 6. log

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