4.1 One-to-One Functions; Inverse Functions. EX) Find the inverse of the following functions. State if the inverse also forms a function or not.

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1 4.1 One-to-One Functions; Inverse Functions Finding Inverses of Functions To find the inverse of function simply switch nd y vlues. Input becomes Output nd Output becomes Input. EX) Find the inverse of the following functions. Stte if the inverse lso forms function or not. A) { (-3, -7), (-, -8), (-1, -1), (0, 0), (1, 1), (, 8), (3, 7)} B) {(-3, 9), (-, 4), (-1, 1), (0, 0), (1, 1), (, 4), (3, 9)} When the inverse of function f is itself function, then f is sid to be one-to-one function. EX) Which emples re one-to-one?

2 EX) For ech function, use the grph to determine whether the function is one-to-one. A function tht is incresing on n intervl I is one-to-one function in I. A function tht is decresing on n intervl I is one-to-one function on I. EX) Inverse: Stte the domin nd rnge of the originl function. Stte the domin nd rnge of the inverse function.

3 EX) Sketch the grph of the inverse on the sme coordinte plne.

4 Finding the Inverse of Function Defined by n Eqution EX) The function 1 f ( ) is one-to-one. Stte the domin nd rnge of 1 f () nd f 1 ( ). HW pgs , 9, 11, 13, 17, 19, 1, 5, 31, 37, 41, 47, 51

5 4. Eponentil Functions Eplortion Grphing Eponentil Functions EX) Grph Domin Rnge f ) ( by hnd nd complete the nlysis. X-Intercept(s) Y-Intercept(s) H.A. V.A. Symmetry End Behvior

6 EX) Grph Domin Rnge 1 f ( ) by hnd nd complete the nlysis. X-Intercept(s) Y-Intercept(s) Symmetry H.A. V.A. End Behvior EX) Grph f ( ) 3 using trnsformtions nd complete the nlysis. Domin Rnge X-Intercept(s) Y-Intercept(s) Symmetry H.A. V.A. End Behvior

7 Wht is e? EX) Grph f ( ) e nd complete the nlysis. Domin Rnge X-Intercept(s) Y-Intercept(s) Symmetry H.A. V.A. End Behvior EX) Grph Domin Rnge 3 f by using trnsformtions nd complete the nlysis. ( ) e X-Intercept(s) Y-Intercept(s) Symmetry H.A. V.A. End Behvior

8 Solving Eponentil Equtions EX) Solve: EX) Solve e ( e ) 3 e EX) Solve 3 EX) HW pg , 13, 15, 17, 19, 3, 7, 31, 35, 37, 39, 43, 45, 47, 51, 55, 57, 61, 65

9 4.3 Logrithmic Functions EX) EX) EX) EX) Find the ect vlue of ech logrithmic epression. b 1 log381 log 8 Domin nd Rnge of Logrithmic Functions

10 EX) Find the domin of ech logrithmic function. f b F log3 log c h log 1 d g log Grphing Logrithmic Functions

11 Grphs of Logrithmic Functions EX) Grph y ln( ) by pplying trnsformtions nd complete the nlysis. Domin Rnge X-Intercept(s) Y-Intercept(s) H.A. V.A. End Behvior Intervls of Incresing Intervls of Decresing

12 EX) Grph y 3log( 1) by pplying trnsformtions nd complete the nlysis. Domin Rnge X-Intercept(s) Y-Intercept(s) H.A. V.A. End Behvior Intervls of Incresing Intervls of Decresing Solving Logrithmic Equtions EX) Be sure to check pprent solutions in the originl eqution nd discrd ny tht re etrneous. Some logrithmic equtions cn be solved by chnging from logrithmic epression to n eponentil epression. b Solve: log 1 3 log Some eponentil equtions cn be solved by chnging the eponentil eqution to logrithmic form. 3 Solve: e 6 EX) EX) Solve: HW pg , 9, 13, 19, 3, 5, 7, 9, 33, 35, 37, 39, 43, 45, 61, 65, 73, 81, 85, 89, 91, 95, 97, 101, 103, 105

13 4.4 Properties of Logrithms Determine the vlue of the following: 1) log 1 ) log EX) Evlute the following: A) log B) log C) kt ln e EX) Rewrite ech epression s sum nd/or difference of logrithms. Epress ll powers s fctors. A) log 1 B) ln 1 3 C) log

14 EX) Rewrite ech of the following s single logrithm. A) log 7 4log 3 B) ln 8 ln C) log log 9 log ( 1) log 5 Common Mistkes: DO NOT DO THESE! log ( M N) log M log N log M log N log M r log log M r log M N EX) Solve A) log 4( 4) log 4() B) log log 7 3

15 EX) Approimte the following. A) log 3 1 B) log 5 C) log 8 EX) Grph y log using your grphing clcultor. EX) Grph y log 1/ using your grphing clcultor. HW pg every other odd 1, 5, 9, 13,, 85

16 4.5 Logrithmic nd Eponentil Equtions Reminder: Logrithms of negtive numbers re not defined so check for etrneous solutions. Solving Logrithmic Equtions EX) Solve log 5 log 9 EX) Solve log 3 4 log 3 5 You cn lso confirm you nswer grphiclly using your clcultor. EX) Solve log 4( 3) log 4( ) 1 Solving Eponentil Equtions EX) Solve EX) Solve EX) Solve 3 7 EX) Solve 8 3 5

17 EX) Solve EX) Solve Solve Using Grphing Clcultor EX) e EX) log 3 log 4 4 HW pg. 37 1, 5, 9, 13, 17, 1, 5, 9, 33, 47, 51, 55

18 4.6 Compound Interest Typiclly interest erned is compound interest, mening interest is pid on previously erned interest.

19 EX) Find the mount A tht results from investing principl P of $000 t n nnul rte r of 8% compounded continuously for time t of 4 yers nd 6 months. EX) A bond cn be redeemed in 10 yers for $1000. How much should you be willing to py for it now if you wnt return of: A) 8% compounded monthly B) 7% compounded continuously EX) Wht nnul rte of interest compounded nnully should you seek if you wnt to double your investment in 5 yers? EX) How long will it tke for n investment to triple in vlue if it erns 5% compounded continuously? HW pg , 5, 9, 11, 17, 3, 5, 35

20 4.7 Growth nd Decy Unnhibited Growth or Decy EX) A colony of bcteri grows ccording to the lw of uninhibited growth ccording to the function, 0.05 e N t t 90, where N is mesured in grms nd t is mesure in dys. EX) A colony of bcteri increses ccording to the lw of uninhibited growth. () If the number of bcteri doubles in 4 hours, find the function tht gives the number of cells in the culture. (b) How long will it tke for the size of the colony to triple? (c) How long will it tke for the popultion to double second time (tht is increse four times)?

21 EX) Trces of burned wood long with ncient stone tools in n rcheologicl dig in Chile were found to contin pproimtely 1.67% of the originl mount of crbon 14. The hlf-life of crbon 14 is 5600 yers. Determine how long go the tree ws cut nd burned. Logistic Model

22 EX) EX) HW pg , 3, 5, 9, 1, 3

23 4.8 Eponentil, Logrithmic, nd Logistic Models EX) Beth is interested in finding function tht eplins the closing price of Hrley Dvidson stock t the end of ech yer. She obtins the dt shown in the tble. A) Using grphing utility, drw sctter digrm with yers s the independent vrible. B) Using grphing utility, fit n eponentil function to the dt. C) Epress the function found in prt (b) in the form kt A A0e. D) Grph the eponentil function found in prt (b) on the sctter digrm. E) Using the solution to prt (b), predict the closing price of Hrley Dvidson stock t the end of 004. F) Interpret the vlue of k found in prt (c).

24 EX) Jodi, meteorologist, is interested in finding function tht eplins the reltion between the height of wether blloon (in kilometers) nd the tmospheric pressure (mesured in millimeters of mercury) on the blloon. She collects the dt shown in the tble. A) Using grphing utility, drw sctter digrm of the dt with tmospheric pressure s the independent vrible. B) It is known tht the reltion between tmospheric pressure nd height follows logrithmic model. Using grphing utility, fit logrithmic function to the dt. C) Drw logrithmic function found in prt (b) on the sctter digrm. D) Use the function found in prt (b) to predict the height of the wether blloon if the tmospheric pressure is 560 millimeters of mercury. EX) The dt in the tble represents the mount of yest biomss in culture fter t hours. A) Using grphing utility, drw sctter digrm of the dt with time s the independent vrible. B) Using grphing utility, fit logistic function to the dt. C) Using grphing utility, grph the function found in prt (b) on the sctter digrm. D) Wht is the predicted crrying cpcity of the culture? E) Use the function found in prt (b) to predict the popultion of the culture t t = 19 hours. HW pg , 7, 9 (let = 0 be 1900)