MATH 43-Precalculus I Chapter 4- (Composition, Inverse), Eponential, Logarithmic Functions I. Composition of a Function/Composite Function A. Definition: Combining of functions that output of one function becomes the input of the net function. B. Notation: ( f g)( ) f [ g( )], read as, composition of f and g of equals to f of g of C. How it works and why should I care about how it works:. Birthday gift inside the bo of another bo, as the composition of a function/composition is like having a function contained inside another function.. Cost of advertising on a TV show 3. Currency conversion from U.S. dollars to Meican pesos, h( ), given: currency echange rate obtained from : http://www.e.com/ucc/convert.cgi) g( ). (US dollars to Canadian dollars), f ( ) 8. 8 (Canadian dollars to Meican pesos) h( ) ( f g)( ) f [ g( )] h( ) f [. ] 8. 8[. ] 0. 56 4. Grading composition
D. Eamples. Given: f ( ) 3, g( ) 5 Find: a. ( f g)( ) b. ( g f )( ) c. ( f g)( 3 ) d. ( g f )( 3 ) e. ( f g)( 3) Solution: a. ( f g)( ) f [ g( )] f [ 5] ( 5) ( 5) 3 0 5 0 3 38 b. ( g f )( ) g[ f ( )] g[ 3] 3 5 c. ( f g)( 3) f [ g( 3)] f [ 3 5] f [ ] ( ) ( ) 3 4 4 3 d. ( g f )( 3) g[ f ( 3)] g[ 3 ( 3) 3] g[ 9 6 3] g[ 6] 6 5 e. ( f g)( 3) f ( 3) g( 3) ( 3 3 3)( 3 5) ( 6)( ) Note: ( f g)( ) ( g f )( ), ( f g)( 3) ( g f )( 3), ( f g)( 3) ( f g)( 3)
4. Given: f ( ), g( ) Find: a. ( f g)( ) b. ( g f )( ) c. ( f g)( ) d. ( g f )( ) e. ( f g)( ) Solution: a. ( f g)( ) Note: the domain, D, of ( f g)( ) b. ( g f )( ) Note: the domain, D, of ( g f )( ) c. ( f g)( ) d. ( g f )( ) e. ( f g)( ) 3. Given: f ( ) 4, g( ) 5 Find: a. ( f g)( ) b. ( g f )( ) c. D of ( f g)( ) Solution: d. ( f g)( ) e. ( g f )( ) f. ( f g)( ) g. ( f f )( ) a. ( f g)( ) e. ( g f )( ) b. ( g f )( ) f. ( f g)( ) c. D of ( f g)( ) g. ( f f )( ) d. ( f g)( ) 3
Note: ( f g)( ) ( f g)( ) and ( f g)( ) ( g f )( ) and ( f g)( ) ( g f )( ) In general, ( f g)( ) ( g f )( ) or f [ g( )] g[ f ( )] Only a special class of functions would have ( f g)( ) ( g f )( ) ;this is the Inverse Function E. Finding Functions That Would Form a Given Composite Function 3 Eample:. Given: h( ) ( f g)( ) f [ g( )] ( 5) 4( 5) 3 Note that ( 5) is a repeated epression/earlier operation 3 if we choose g( ) 5 f ( ) 4 3 Note: usually equate earlier operation to function on the inside, g( ), in this case 3 ( f g)( ) f [ g( )] f [ 5] ( 5) 4( 5) 3 There are other pairs of functions of f and g that would also work. For instance, note that is a repeated epression/earlier operation. if we choose g( ), f ( ) ( f g)( ) f [ g( )]. Given: h( ) ( f g)( ) f [ g( )] 4, Find: f ( ), g( ) Result : g( ) f ( ) Result : g( ) f ( ) 4
Check: h( ) f [ g( )] h( ) f [ g( )] 3. Given: h( ) = f [ g( )], Find: f ( ), g( ) Result : g( ) f ( ) Result : g( ) f ( ) Check: h( ) f [ g( )] 5
II. Inverse Function A. Definition: Recall, in general, ( f g)( ) ( g f )( ) or f [ g( )] g[ f ( )] Only in a very special class of functions, such that the composite functions, ( f g)( ) ( g f )( ) or f [ g( )] g[ f ( )], this is the Inverse Function. B. Notation: f ( ), the magic wand, "un-doing" C. Generating an Inverse Function f ( ) 5 to generate an inverse function of the original function, we need to y 5 write the function in this form y 5 swap the and y 5 y then solve for y 5 y this is the inverse function of the original f ( ), f ( ) 5 Note that in the notation for the inverse function, f is not an eponent. ( ), the - 6
D. Acid Test for Inverse Function If given f ( ) 5, g( ) 5 Determine: f [ g( )] and g[ f ( )] Therefore, If f [ g( )] g[ f ( )] then, g( ) f ( ) f ( ) g ( ) or, ( f f )( ) and ( f f )( ) This is also equivalent to the concept of "Undo" in programming...or equivalent to having the magic wand that can undo the moment when the vase was broken into pieces. E. Compare the Domain and Range of a Function and Its Inverse Eample: Given: f ( ) 6 R of f ( ) Find: D & R of f ( ) and D & Domain of the function is equal to the Range of its inverse function whereas the Range of the function is equal to the Domain of its inverse function. F. Graphical Relationship between a Function and Its Inverse : (Swapping between D and R roles of f ( ) and f ( ) ) 7
Eample: f ( ) 3 g( ) 3 y_ y_ - -5 0-3 - A function and its inverse function are reflection of one another across the line y. Horizontal line test of the original function as a way to determine if its inverse is a function. Now, let's generate the inverse function of another function: Eample: f ( ) 4 g( ) 4 We can see that when a function and its inverse are both of functional relationship, there is the one to one idea. That usually occurs when the function is continuously increasing or decreasing. 8
S. Nunamaekr G. The graphing calculator (TI-83) http://www.mathbits.com/mathbits/tisection/algebra/inverse.htm. Using graphing calculator (TI-83)to quickly determine if the inverse of your graph is a function using the horizontal line test.. Using TI-83, and the fact that f [ f ( )] f [ f ( )], so graphing y 3, y3 y( y), if y and y are inverse of each other, the composition of y( y) would yield the identity function, y. This would verify if one is the inverse of the other. 3. Using TI-83's table feature to verify the swapping of and y values between functions and their inverses. H. Important Facts about Inverses. If f is one-to-one, then f eists. The domain of f is equal to the range of f, and the range of f is equal to the domain of f. 3. If the point (a, b) lies on the graph of f, then (b, a ) lies on the graph of the f, so the graph of f and the graph of f are reflections of each other across the line y. 4. To find the equation for f, replace f ( ) with y, interchange and y, and solve for y. This gives f ( ). 9
III. Eponential Function Eponential Functions and Graphs A. Eponential Functions The function f ( ) a, where is a real number, a 0, and a, is called the eponential function, base a. Requiring the base to be positive would help to avoid the comple numbers that would occur by taking even roots of negative numbers. (E., ( ), which is not a real number.) The restriction a is made to eclude the constant function f ( ). Eample: f ( ) 5, f ( ) ( ), 6 f ( ) ( 4. 75) *The variable in an eponential function is in the eponent. B. Graphing an eponential function. Plotting points. Graphing calculator * Try f ( ) ( ) 0
C. Application i Eample: Compound Interest Formula, A P( ) n r for n compoundings per year: or A P( ) n for continuous compounding: A Pe rt. a total of $,000 is invested at an annual rate of 9%. Find the balance after 5 years if it is compounded quarterly: r nt. ( ) A P( ), 000( 0 09 4 5 ) 8, 76. n 4. a total of $,000 is invested at an annual rate of 9%. Find the balance after 5 years if it is compounded continuously: rt 0. 09( 5) A Pe, 000e 8, 89. 75 continuous compounding yields interest: 889.75-876.=93.64 D. The Number e. 788884... n A( n) ( ), as the n gets larger and larger, the function value n gets closer to e. Its decimal representation does not terminate or repeat; it is irrational. In 74, Leonard Euler named this number e. You can use the e key on a graphing calculator to find values of the eponential function f ( ) e. Eample: Find e 3 0 3, e., e 0, 00e 5. 8, e nt nt
E. Eponents Evaluate eponential epressions.. For any positive integer n, a n a a a a a n times such that a is the base and n is the eponent Eample: a 3 a a a 3 3 33 3 = 7. For any nonzero real number a and any integer n, a 0 And a n a n 3. In an multiplication problem, the numbers or epressions that are multiplied are called factors. If a b c, then a and b are factors of c. Eamples: a. 6 0 b. ( ) 0 c. ( ) 3 d. 3 = e. 4 = 4. Properties of Eponents a a a m n m n a a m n a ( mn) such that a 0 ( a ) ( ab) a m n mn a b m m m a ( ) b m m a such that b 0 m b
Eamples: a. ( ) 5 b. y 5 y c. y y 6 3 d. ( ) 3 4 e. m 5 5 m 0 8 7 4a b c f. ( ) 6 3 5 3a b c 5 = F. Fractional Eponents Definition.: a radicand. n n a such that n is called the inde and a is the m n m Definition : a a ( a ) n n m * m is an integer, n is a positive integer, and a is a real number. If n is even, a 0. Eamples 3 3 3 3 9 3 3 4 4 3 3 4 3 3 4 4 64 ( ) 44 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 ( )( ) ( )( ) ( ) 3 3 3 ( 5 ) 5 3 5 5 8 8 3 3 3 3 3 3 3 3 8 8 3 ( ) ( )( ) 3
IV. Logarithmic Functions Logarithmic Functions and Graphs A. Logarithm Logarithmic, or logarithm, functions are inverse of eponential functions and have many applications. Eample: Convert each to a logarithmic equation: 6 >log 6 (the logarithm, base, of 6 ) (the power to which we raise to get 6) 0 3 0. 00 log 0. 00 3 0 e t 70 log 70 t e Eample: Find each of the following logarithms: log 0 0, 000 log. 0 0 0 log 8 log 9 3 log 6 log 8 8 * log a 0 and log a a, for any logarithmic base a 4
B. Common Logarithm and Natural Logarithm. Common logarithms are of base-0. The abbreviation log, with no base written, is used to represent the common logarithms, or base 0 logarithms. Eample: log 9 means log 0 9. Natural Logarithms log 00 = log 0 00 Logarithms, base e, are called natural logarithms. The natural logarithm's abbreviation is ln. Eample: ln 3 means log e 3 C. Changing Logarithmic Bases log b M log log 0 0 M b ln M ln b Eample: log 4 log0 56. 4084 56 4 log 4 0. 6006 0 ln log 56 56 55458. 4 4 ln 4 3869. 5
D. Properties of Logarithms. b b b y y 3 4 34 Eample: 8. b b y b y Eample: 5 3 53 4 3. ( b ) b y y 3 4 Eample: ( ) 4096 4. log y log log y b b b Eample: log 43 log ( 7 9) log 7 log 9 3 5 3 3 3 3 5. log ( ) log log y b b b y 6 Eample: log ( ) log 6 log 8 4 3 8 n 6. log ( ) nlog b 7. log b 0 b 3 Eample: log ( 4 ) 3log 4 3 6 Eample: log 8 0 6
8. log b b Eample: log 8 8 n 9. log ( b ) n b 4 Eample: log ( ) 4 0. f ( ) log ln, 0 e is the natural logarithmic function. ln e. ln ln e e 3 ln 0 4. ln( uv) ln u ln v E. Try these: Epress each as a sum, difference, or multiple of logarithms. log 5 33. log 4 5 3. log 3 9 4. log 6 5. log 5 5 6. log 7 7 7. log 5( ) 5 8. log 7 3 y 4 9. log 3( ) 8 3 0. log 0 5 y. ln 3 5 7 3. log b 75 7
Epress each as the logarithm of a single quantity. log a log c. log 9 log 3 b b 5 5 3. log e 3log n 4. e log b a log 5 b 5. 3 ln ( ) 6. 3ln ln y 4ln z 8
VI. Solving Eponential and Logarithmic Equations A. To solve an eponential equation, first isolate the eponential epression, then take the logarithm of both sides and solve for the variable. Eample: Solving 3 5 64 5 Solving e 7 ln e ln7 ln 7 946. Solving e 5 60 e 55 ln e ln55 ln 55 4. 0 Solving e 3e 0 ( e )( e ) 0 e, e if e ln e ln ln. 693 if e ln e ln 0 B. To solve a logarithmic equation, rewrite the equation in eponential form (eponentiating) and solve for the variable. ln Eample: Solving ln e e e 7. 389 Solving 5 4 ln ln ln ln e e. 607 Solving ln 3 e ln3 4 ln 3 e e 3 e. 46 3 Solving ln ln( ) ln e e e e e e e ( e) e e e 9
C. Application Eample: You have deposited $500 in an account that pays 6.75% interest, compounded continuously. a. How long will it take your money to double? b. How long will it take your money to triple? rt a. for continuous compounding, A Pe 500e. 0675 or P( t) P e kt. 0 =500e 0675 To find the time required to double, A 000 500e. 0675. or P( t) 000 500e 0675 t t t t. 0675 e t. 0675t ln ln e ln. 0675t ln t 0 7. 0675. b. To find the time required to triple, A 500 500e. 0675. or P( t) 500 500e 0675 t t. 0675 e t. 0675t ln3 3 ln e ln 3. 0675t ln3 t 6 8. 0675. Doubling Time, T Eponential Growth Rate, k ln, Tripling Time, T k ln T or k ln3 T3 3 ln3 k 0
V. Eponential and Logarithmic Models A. The five most common types of mathematical models involving eponential functions and logarithmic functions: b kt. Eponential growth: y ae, b 0 or P( t) P e, k 0 0 * b or k is the eponential growth rate. b kt. Eponential decay: y ae, b 0 or P( t) P e, k 0 0 * b or k is the eponential decay rate (E. radioactive substance) Half-life: half of the radioactive substance will cease to be radioactive within that period of time. 3. Gaussian model: y ae ( b) c This type of model is used in probability and statistics to represent populations that are normally distributed. The standard normal distribution takes the form y The graph of a Gaussian model is bell-shaped curve. 4. Logistics growth model or logistic function: y (y = pop. size, = time) a be ( c) d a or P( t ) this function increases toward a limiting be kt value as t. Thus, the horizontal asymptote of y a or, P( t) approaches a as t. So, a is the limiting value in this model of limited growth. e
Some population initially have rapid growth, followed by a declining rate of growth. This type of growth pattern is of logistics curve; it is also called a sigmoidal curve. Eample: bacteria culture or spread of an epidemic Try: In a town where population is 4500, a disease creates an epidemic. The number of people N infected t days after the disease has 4500 begun is given by the function: N ( t). t 9. 9e 0 6 a. Graph the function b. How many are initially infected with the disease (t 0)? c. Find the number infected after days, 5days, 8 days? d. Using this model, can you say whether all 4500 people will ever be infected? 5. Logarithmic model: y ln( a b), y log ( a b) E.: On the Richter scale, the magnitude of R of an earthquake of I intensity I: R log 0 I0 B. Take a look at the basic shapes of these graphs:. y e. y e 3. y e 0 4. y e 5. y ln 6. y log 0