Chapter 9 Vocabulary Check

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1 9 CHAPTER 9 Eponential and Logarithmic Functions Find the inverse function of each one-to-one function. See Section f = f = - CONCEPT EXTENSIONS The formula = 0 e kt gives the population size of a population that eperiences an annual rate of population growth k (given as a decimal). In this formula, t is time in ears and 0 is the initial population at time 0. Use this formula to solve Eercises 69 and In 00, the population of Michigan was approimatel 9,99,000 and decreasing according to the formula = 0 e t. Assume that the population continues to decrease according to the given formula and predict how man ears after which the population of Michigan will be 9,00,000. (Hint: Let 0 = 9,99,000; = 9,00,000, and solve for t.) (Source: U.S. Bureau of the Census) 70. In 00, the population of Illinois was approimatel,80,000 and increasing according to the formula = 0 e 0.00t. Assume that the population continues to increase according to the given formula and predict how man ears after which the population of Illinois will be,00,000. (See the Hint for Eercise 69.) (Source: U.S. Bureau of the Census) 7. When solving a logarithmic equation, eplain wh ou must check possible solutions in the original equation. 7. Solve = 9 b taking the common logarithm of both sides of the equation. Net, solve this equation b taking the natural logarithm of both sides. Compare our solutions. Are the the same? Wh or wh not? Use a graphing calculator to solve each equation. For eample, to solve Eercise 7, let Y = e 0. and Y = 8 and graph the equations. The -value of the point of intersection is the solution. Round all solutions to two decimal places. 7. e 0. = = 7 7. log = ln = Check Eercise. 78. Check Eercise. 79. Check Eercise. 80. Check Eercise. Chapter 9 Vocabular Check Fill in each blank with one of the words or phrases listed below. Some words or phrases ma be used more than once. inverse common composition smmetric eponential vertical logarithmic natural half-life horizontal. For a one-to-one function, we can find its function b switching the coordinates of the ordered pairs of the function.. The of functions f and g is f g = f g.. A function of the form f = b is called a(n) function if b 7 0, b is not, and is a real number.. The graphs of f and f - are about the line =.. logarithms are logarithms to base e. 6. logarithms are logarithms to base To see whether a graph is the graph of a one-to-one function, appl the line test to see whether it is a function and then appl the line test to see whether it is a one-to-one function. 8. A(n) function is a function that can be defined b f = log b where is a positive real number, b is a constant positive real number, and b is not. 9. is the amount of time it takes for half of the amount of a substance to deca. 0. A quantit that grows or decas b the same percent at regular time periods is said to have growth or deca.

2 Chapter 9 Highlights 9 Chapter 9 Highlights DEFINITIONS AND CONCEPTS Section 9. The Algebra of Functions; Composite Functions EXAMPLES Algebra of Functions Sum f + g = f + g Difference Product Quotient f - g = f - g f # g = f # g a f g f b =, g 0 g Composite Functions The notation f g means f composed with g. f g = f g g f = gf Section 9. If f is a function, then f is a one-to-one function onl if each -value (output) corresponds to onl one -value (input). Horizontal Line Test If ever horizontal line intersects the graph of a function at most once, then the function is a one-to-one function. If f = 7 and g = +, f + g = f + g = f - g = f - g = = f # g = f # g = 7 + = a f f b = g g = 7 + If f = + and g = -, find f g. Inverse Functions f g = f g = f - = - + = Determine whether each graph is a one-to-one function. A B C Graphs A and C pass the vertical line test, so onl these are graphs of functions. Of graphs A and C, onl graph A passes the horizontal line test, so onl graph A is the graph of a oneto-one function. The inverse of a one-to-one function f is the one-to-one function f - that is the set of all ordered pairs (b, a) such that (a, b) belongs to f. To Find the Inverse of a One-to-One Function f() Step. Replace f () with. Step. Interchange and. Step. Solve for. Step. Replace with f -. Find the inverse of f = + 7. = + 7 Replace f () with. = + 7 Interchange and. = - 7 Solve for. = - 7 f - = - 7 Replace with f -. The inverse of f = + 7 is f - = - 7.

3 9 CHAPTER 9 Eponential and Logarithmic Functions DEFINITIONS AND CONCEPTS EXAMPLES Section 9. Eponential Functions A function of the form f = b is an eponential function, where b 7 0, b, and is a real number. Graph the eponential function = Uniqueness of b If b 7 0 and b, then b = b is equivalent to =. Solve + = 8. + = Write 8 as. + = Use the uniqueness of b. = - Subtract from both sides. Section 9. Eponential Growth and Deca Functions A quantit that grows or decas b the same percent at regular time periods is said to have eponential growth or eponential deca. Eponential Growth n initial amount = C + r number of time intervals + r is growth factor r is growth rate (often a percent) Eponential Deca number of time intervals initial n amount = C - r n nn - r is deca factor r is deca rate (often a percent) n Section 9. A cit has a current population of 7,000 that has been increasing at a rate of % per ear. At this rate, find the cit s population in 0 ears. = C + r = 7, ,86. In 0 ears, the predicted population of the cit is 66,86. A cit has a current population of 7,000 that has been decreasing at a rate of % per ear. At this rate, find the cit s population in 0 ears. = C - r = 7, ,0.9 In 0 ears, predicted population of the cit is 0,0. Logarithmic Functions Logarithmic Definition If b 7 0 and b, then = log b means = b for an positive number and real number. Properties of Logarithms If b is a real number, b 7 0, and b, then log b = 0, log b b =, b log b = Logarithmic Corresponding Form Eponential Statement log = = log 9 = 9 / = log = 0, log 7 7 =, log 6 = 6

4 Chapter 9 Highlights 9 DEFINITIONS AND CONCEPTS EXAMPLES Section 9. Logarithmic Functions (continued) Logarithmic Function If b 7 0 and b, then a logarithmic function is a function that can be defined as f = log b The domain of f is the set of positive real numbers, and the range of f is the set of real numbers. Graph = log. Write = log as =. Plot the ordered pair solutions listed in the table and connect them with a smooth curve Section 9.6 Let,, and b be positive numbers and b. Product Propert log b = log b + log b Quotient Propert log b = log b - log b Power Propert Properties of Logarithms Write as a single logarithm. log 6 + log - log + = log 6 + log - log + Power propert = log 6 # - log + Product propert 6 = log + Quotient propert log b r = r log b Section 9.7 Common Logarithms, Natural Logarithms, and Change of Base Common Logarithms log means log 0 Natural Logarithms log = log ln 7 = log e 7.99 ln means log e Continuousl Compounded Interest Formula A = Pe rt where r is the annual interest rate for P dollars invested for t ears. Section 9.8 Find the amount in an account at the end of ears if $000 is invested at an interest rate of % compounded continuousl. Here, t = ears, P = +000, and r = 0.0. A = Pe rt = 000e Eponential and Logarithmic Equations and Problem Solving Logarithm Propert of Equalit Let log b a and log b c be real numbers and b. Then log b a = log b c is equivalent to a = c Solve =. log = log Logarithm propert of equalit log = log Power propert = log log Divide both sides b log..9 Use a calculator.

5 96 CHAPTER 9 Eponential and Logarithmic Functions Chapter 9 Review (9.) If f = - and g = +, find. f + g. f - g. f # g. a g f b If f = -, g = +, and h = -, find each composition.. f g 6. g f 7. h g 8. f f f g- 0. h h (9.) Determine whether each function is a one-to-one function. If it is one-to-one, list the elements of its inverse.. h = -9,, 6, 8, -,,, 6. f = -,, 0,,,,, -66. Find an equation defining the inverse function of the given oneto-one function.. f = - 9. f = + 8. f = 6 +. f = -. f = - 6. f = + U.S. Region (Input) Northeast Midwest South West Rank in Housing Starts for 009 (Output) 7. g = r = - Graph each one-to-one function and its inverse on the same set of aes. 9. f = f = -. Shape (Input) Square Triangle Parallelogram Rectangle Number of Sides (Output) Given that f = + is a one-to-one function, find the following.. a. f(7) b. f - 6. a. f - b. f - Determine whether each function is a one-to-one function (9.) Solve each equation for.. = 6. = 9. =. = = = Graph each eponential function. 7. = 8. = a b 9. = - 0. = + Use the formula A = Pa + r nt n b to solve the interest problems. In this formula, A = amount accrued or owed P = principal invested or loaned r = rate of interest n = number of compounding periods per ear t = time in ears. Find the amount accrued if $600 is invested at 9% interest compounded semiannuall for 7 ears.. A total of $800 is invested in a 7% certificate of deposit for which interest is compounded quarterl. Find the value that this certificate will have at the end of ears.

6 Chapter 9 Review 97 (9.) Solve. Round each answer to the nearest whole.. The cit of Henderson, Nevada, has been growing at a rate of.% per ear since the ear 000. If the population of Henderson was 79,087 in 000 and this rate continues, predict the cit s population in 00.. The cit of Raleigh, North Carolina, has been growing at a rate of.% per ear since the ear 000. If the population of Raleigh was 87,70 in 000 and this rate continues, predict the cit s population in 08.. A summer camp tournament starts with 0 plaers. After each round, half the plaers are eliminated. How man plaers remain after 7 rounds? 6. The bear population in a certain national park is decreasing b % each ear. If this rate continues, and there is currentl an estimated bear population of 80, find the bear population in 6 ears. (9.) Write each equation with logarithmic notation = = 6 Write each logarithmic equation with eponential notation. 9. log / 6 = - 0. log = Solve for.. log = -. log =. log =. log 6 =. log = 6. log = 7. log = 8. log 9 = 9. log - = 60. log + = 6. log - = 6. log = Graph each pair of equations on the same coordinate sstem. 6. = and = log 6. = a b and = log / (9.6) Write each of the following as single logarithms. 6. log 8 + log 66. log 6 + log 67. log 7 - log log 8 - log 69. log 8 + log - log log + log - log 7. log - log + + log 7. log - log + log + Use properties of logarithms to write each epression as a sum or difference of multiples of logarithms. 7. log + 7. log + 7. log z 76. log z 7 If log b = 0.6 and log b = 0.8, find the following. 77. log b log b (9.7) Use a calculator to approimate the logarithm to four decimal places. 79. log log ln. 8. ln.6 Find the eact value. 8. log log 0 8. ln 86. ln e e Solve each equation for. 87. ln = 88. ln = ln - = ln + = Use the formula ln I = -k to solve the radiation problem in 9 I 0 and 9. In this formula, = depth in millimeters I = intensit of radiation I 0 = initial intensit k = a constant measure dependent on the material Round answers to two decimal places. 9. Find the depth at which the intensit of the radiation passing through a lead shield is reduced to % of the original intensit if the value of k is.. 9. If k is., find the depth at which % of the original radiation will penetrate. Approimate the logarithm to four decimal places. 9. log.6 9. log Use the formula A = Pe rt to solve the interest problems in which interest is compounded continuousl. In this formula, A = amount accrued or owed P = principal invested or loaned r = rate of interest t = time in ears 9. Bank of New York offers a -ear, % continuousl compounded investment option. Find the amount accrued if $0 is invested. 96. Find the amount to which a $90 investment grows if it is invested at % compounded continuousl for ears. (9.8) Solve each eponential equation for. Give an eact solution and approimate the solution to four decimal places. 97. = =

7 98 CHAPTER 9 Eponential and Logarithmic Functions = = = = 0. - = 0. + = Solve the equation for. 0. log + log = 06. log + log 0 = 07. log - log + = 08. -log log 6 = 09. log + log - = 0. log - 8 = Use the formula = 0 e kt to solve the population growth problems. In this formula, = size of population 0 = initial count of population k = rate of growth written as a decimal t = time Round each answer to the nearest tenth.. In 987, the population of California condors was onl 7 birds. The were all brought in from the wild and an intensive breeding program was instituted. If we assume a earl growth rate of.%, how long will it take the condor population to reach 7 California condors? (Source: California Department of Fish and Game). France is eperiencing an annual growth rate of 0.%. In 00, the population of France was approimatel 6,8,000. How long will it take for the population to reach 70,000,000? Round to the nearest tenth. (Source: Population Reference Bureau). In 00, the population of Australia was approimatel,600,000. How long will it take Australia to double its population if its growth rate is 0.7% annuall? Round to the nearest tenth. (Source: Population Reference Bureau). Israel s population is increasing in size at a rate of.6% per ear. How long will it take for its population of 7,76,00 to double in size? Round to the nearest tenth. (Source: Population Reference Bureau) Use the compound interest equation A = Pa + r nt n b to solve the following. (See the directions for Eercises and for an eplanation of this formula.) Round answers to the nearest tenth.. Find how long it will take a $000 investment to grow to $0,000 if it is invested at 8% interest compounded quarterl. 6. An investment of $6000 has grown to $0,000 while the mone was invested at 6% interest compounded monthl. Find how long it was invested. Use a graphing calculator to solve each equation. Round all solutions to two decimal places. 7. e = = 7 MIXED REVIEW Solve each equation. 9. = = =. 9 - = 7. log =. log =. log - = 6. log - = 7. ln = log + log 0 = 9. ln - ln = 0. log 6 - log = Chapter 9 Test If f = and g = -, find the following.. f # g. f - g If f =, g = - 7, and h = - 6 +, find the following.. f h0. g f. g h On the same set of aes, graph the given one-to-one function and its inverse. 6. f = 7 - Determine whether the given graph is the graph of a one-to-one function

8 Chapter 9 Test 99 Determine whether each function is one-to-one. If it is one-toone, find an equation or a set of ordered pairs that defines the inverse function of the given function. 9. f = 6-0. f = 0, 0,,, -, 6. Word (Input) Dog Cat House Desk Circle First Letter of d c h d c Word (Output) Use the properties of logarithms to write each epression as a single logarithm.. log 6 + log. log + log - log +. Write the epression log 6 as the sum or difference of multiples of logarithms.. If log b = 0.79 and log b =.6, find the value of log b. 6. Approimate log 7 8 to four decimal places. 7. Solve 8 - = for. Give an eact solution Solve + = for. Give an eact solution and approimate the solution to four decimal places. Solve each logarithmic equation for. Give an eact solution. 9. log = - Use the formula A = Pa + r nt n b to solve Eercises Find the amount in an account if $000 is invested for ears at 9% interest compounded monthl. 8. Find how long it will take $000 to grow to $000 if the mone is invested at 7% interest compounded semiannuall. Round to the nearest whole. 9. Suppose ou have $000 to invest. Which investment, rounded to the nearest dollar, ields the greater return over 0 ears: 6.% compounded semiannuall or 6% compounded monthl? How much more is ielded b the better investment? Solve. Round answers to the nearest whole. 0. Suppose a cit with population of 0,000 has been decreasing at a rate of % per ear. If this rate continues, predict the population of the cit in 0 ears.. The prairie dog population of the Grand Forks area now stands at 7,000 animals. If the population is growing at a rate of.6% annuall, how man prairie dogs will there be in that area ears from now?. In an attempt to save an endangered species of wood duck, naturalists would like to increase the wood duck population from 00 to 000 ducks. If the annual population growth rate is 6.%, how long will it take the naturalists to reach their goal? Round to the nearest whole ear. The reliabilit of a new model of CD plaer can be described b the eponential function Rt =.7 -/t, where the reliabilit R is the probabilit (as a decimal) that the CD plaer is still working t ears after it is manufactured. Round answers to the nearest hundredth. Then write our answers as percents.. What is the probabilit that the CD plaer will still work half a ear after it is manufactured? 0. lne =. log 8 - =. log + log =. log + - log - =. Solve ln + 7 =. accurate to four decimal places.. What is the probabilit that the CD plaer will still work ears after it is manufactured?. Graph = a b Graph the functions = and = log on the same coordinate sstem.

9 600 CHAPTER 9 Eponential and Logarithmic Functions Chapter 9 Cumulative Review. Multipl. a. -8- b. - 6 c d. 0 - e. a 0 ba- b f. 7-- g Solve: - = +. Graph =.. Find the equation of a line through -, 6 and perpendicular to f = - +. Write the equation using function notation.. Solve the sstem. μ - - z = z = z = 6. Line l and line m are parallel lines cut b transversal t. Find the values of and. t ( 0) 7. Use the quotient rule to simplif. a. 7 8 b. c. 06 l m d. 0 7 z 8 z 7 8. Use the power rules to simplif the following. Use positive eponents to write all results. a. a b. a - b c. a a b b e. a - b c - - d. a - b - 9. For the ICL Production Compan, the rational function.6 + 0,000 C = describes the compan s cost per disc for pressing compact discs. Find the cost per disc for pressing: a. 00 compact discs b. 000 compact discs 0. Multipl. a. - b. a + ba - b c Add or subtract. a. + c b. d. 7z + 7z - +. Perform the indicated operations and simplif if possible Divide b -.. Simplif each comple fraction. a. a a - 0 c Solve: b. - + = - 6. Divide - 8 b -. + a a a + - a - 7. Steve Deitmer takes times as long to go 7 miles upstream in his boat as he does to return. If the boat cruises at 0 mph in still water, what is the speed of the current? 8. Use snthetic division to divide: , - 9. Simplif the following epressions. a. 8 b. - c. - d. -8 e Solve a + = a a + a + 7a + 0. Use rational eponents to write as a single radical. a. # b. c. #. Suppose that varies directl as. If = when =, find the constant of variation and the direct variation equation.

10 Chapter 9 Cumulative Review 60. Multipl. a. + 0 b c d. - e. - + f Solve = Rationalize the denominator. A 7 m n 8. Solve Find the length of the unknown side of the triangle.. Find each root. Assume that all variables represent nonnegative real numbers. a. 7 b. 9-7 c. A 6 d. e in.. Graph F = - +. in.. Rationalize the denominator of 6. Multipl. a. a / a / - a 7/ b. / - / + 7. Solve - = Solve = 0 b completing the square. 0. Add or subtract as indicated. a. 0 + b. A Use the quotient rule to divide and simplif if possible. Assume that all variables represent positive numbers. a. 6 c. 8a b 0 b b. 08a 6. Find the following powers of i. a. i 8 b. i c. i d. i - 7. If f = - and g = -, find a. f + g b. f - g c. f # g f d. a g b 8. Solve = 0 b completing the square. 9. Find an equation of the inverse of f = Solve b using the quadratic formula. a - b =. Find the value of each logarithmic epression. a. log 6 b. log 0 c. log 0 9. Graph f = Find the verte and the ais of smmetr.

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