Sections 1.3, 7.1, and 9.2: Properties of Exponents and Radical Notation

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1 Sections., 7., nd 9.: Properties of Eponents nd Rdicl Nottion Let p nd q be rtionl numbers. For ll rel numbers nd b for which the epressions re rel numbers, the following properties hold. i = + p q p q p = nd p p b = b p p p = p q 5 ( = p) q p q = pq 6 ( ) p p p b = b 7 p = b b p p 8 The n th root of rel number is b if n is not rel number. n b = nd the principl n th root is denoted n. If < nd n is even then 9 If m nd n re positive integers with m n in lowest terms, then ) = n n m b) n n m n = = ( ) If < nd n is even then m m n is not rel number. Pge of

2 The Allownce Problem Would you prefer monthly llownce of : ) \$ or b) to strt with cents nd double it ech dy, i.e. on dy you would receive \$., dy - \$.8, etc, or c) to strt with \$ nd hve the llownce increse by \$5 ech dy. How much will you hve t the end of thirty dys? A B C Dy Amount Dy Amount Dy Amount Pge of

3 The Ppercup Problem Strt with one cup. Proceed to dd cups for ech turn. f ( ) 5 6 Strt with one cup. Double the number of cups for ech turn. g( ) 5 Pge of

4 Section (9.) An eponentil function is Function function whose eqution cn f ( ) = C be put into the form f() C =, where C, >,nd. The constnt b is clled the bse. f ( ) = C = = Grph Domin Rnge y- intercept Use your grphing clcultor to complete the following chrt. How does the vlue of C ffect the grph? f ( ) = 5 C = = f ( ) = ( 7) C = = Eponentil growth: f ( ) = C = = Nturl Bse e: e = (irrtionl number) Nturl Eponentil Function: Eponentil decy: f ( ) = 5 C = = f ( ) = C = = ( ) ( ) f = C = = Pge of

5 Sketch the grph of f ( ) = nd g= ( ). Use your clcultor to verify your nswer. f() g() For n eponentil function of the form y= f( ) = b, if the vlue of the independent vrible increses by, then the vlue of the dependent vrible is multiplied by the bse b. For liner function y = f( ) = m+ b, if the vlue of the independent vrible increses by, then the vlue of the dependent vrible chnges by the slope m. Pge 5 of

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7 Compound Interest: A person invests \$5 in n ccount t 5% interest compounded nnully. Let A= f () t represent the vlue (in dollrs) of the ccount fter t yers or ny frction thereof.. Find nd interpret the following within the contet of the sitution. Use the tble below if necessry. f(), f(), nd f(). b. Complete the tble t A= f() t 5 5 5(.5) 5(.5) 5(.5) + = + = = + = + = = 5(.5) 5(.5)(.5) 5(.5)(.5) 5(.5) 5 c. Find n eqution for f. Pge 7 of

8 Composite & Inverse Functions (9.) ( ) The composite of g nd f is given by ( g f)( ) = g f ( ), nd represents new function whose nme is g f. Suppose f ( ) = nd g( ) =. Find ) g f b) ( f g) c) ( f g)() Pge 8 of

9 Complete the tbles below. f ( ) = g( ) = - - h ( ) = k( ) = Informlly, two functions re inverse functions if they undo ech other. f nd g re inverses of ech other, nd h nd k re inverses of ech other. f is clled the inverse of f (or f inverse) nd undoes f. domin - - f f rnge -8-8 f sends its inputs to the corresponding outputs. Emple: f () = 8 f sends the outputs of f to the corresponding inputs of f. Emple: f ( ) =. Complete the tble below. f() f ( ) Pge 9 of

10 . Describe the mening of f in terms of inputs nd outputs for the function f. How do you find the inverse of function? Tke the function f ( ) = + 5. Describe wht is being done to the inputs of the function f.. How does the inverse function, f, undo this? Describe the process.. Wht is f? Pge of

11 Given f ( ) =, grph f ( ) nd f ( ) y = Pge of

12 Introduction to Logrithms nd Logrithmic Functions (9.) A common logrithm is logrithm with bse. We represent the log ( ) s log( ). A nturl logrithm is logrithm with bse e. We represent log e ( ) s ln ( ). For >, ln( ) = c nd c e = re equivlent. Wht is the inverse of n eponentil function of the form f ( ) = b? Wht is the inverse of logrithmic function of the form g ( ) = log b? A nturl logrithmic function is function tht cn be put in the form g( ) = ln where the input vlues for re positive numbers. g ( ) = e. Find the inverse of the given function.. g= ( ) 5 b. h ( ) = log Pge of

13 Properties of Logrithms:.. Evlute the logrithmic function t the given vlue. log = i. ( ) ii. log7 ( 7 ) = iii. log ( ) = b. Wht is the vlue of log b b? Eplin... Evlute the following logrithmic functions t the indicted vlues. log = i. ( ) 5 log = ii. ( ) 6 iii. log ( ) = b. Wht is the vlue of log ( ) b? Eplin.. ln( ) = becuse. 5. ln( e ) = becuse. 6. Chnge of Bse Formul: log Putting it ll together:. Find n eqution for h b. Stte the domin nd rnge of h. c. Evlute h( ) d. Find n eqution for e. Sketch the grph of h. log ln = = log ln h. Lbel two points on the grph. Pge of

14 f. Evlute h (6). Solving Eponentil & Logrithmic Equtions (9.5): Eponentil Equtions. Solve for the eponentil epression.. Tke logrithm of ech side.. Apply the Power Property of logrithms.. Solve the resulting eqution Logrithmic Equtions. Solve for the logrithm in the eqution.. Rewrite in eponentil form.. Solve the resulting eqution.. Check for etrneous solutions. Pge of