Assuming that all items produced are sold, find the cost C as a function of the price p.

Transcription

1 Math Reviewing Chapter 5 Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. For the given functions f and g, find the requested composite function value. 1) f(x) = x - 6 x, g(x) = x2 + 9; Find (g f)(-2). 1) Find functions f and g so that f g = H. 1 2) H(x) = x2-9 2) For the given functions f and g, find the requested composite function. 3) f(x) = 3 x + 8, g(x) = 5 ; Find (f g)(x). 3) 6x Find the domain of the composite function f g. x 20 4) f(x) = ; g(x) = x + 10 x + 2 4) 5) The surface area S (in square inches) of a cylindrical pipe with length 12 inches is given by S(r) = 2 r r, where r is the radius of the piston (in inches). If the radius is increasing 5) with time t (in minutes) according to the formula r(t) = 1 6 t2, t 0, find the surface area S of the pipe as a function of the time t. Find the domain of the composite function f g. 6) f(x) = -72 x ; g(x) = 8 x + 8 6) The function f is one-to-one. Find its inverse. 7) f(x) = 5x2 + 2, x 0 7) 8) The price p of a certain product and the quantity sold x obey the demand equation p = x + 200, 0 x 300. Suppose that the cost C of producing x units is C = x Assuming that all items produced are sold, find the cost C as a function of the price p. 8) The function f is one-to-one. Find its inverse. 9) f(x) = -8x + 6-9x - 6 9) 1

2 10) The weight W of a bird's brain (in ounces) is related to the volume V of the bird's skull (in 10) cubic ounces) through the function W(V) = V (a) Express the skull volume V as a function of brain weight W. (b) Predict the skull volume of a bird whose brain weighs 2 oz. For the given functions f and g, find the requested composite function. 11) f(x) = 7 x - 5, g(x) = 4 ; Find (f g)(x). 11) 3x Find the domain of the composite function f g. 12) f(x) = 5 ; g(x) = x ) x - 7 For the given functions f and g, find the requested composite function. 13) f(x) = x + 9, g(x) = 8x - 13; Find (f g)(x). 13) Find functions f and g so that f g = H. 14) H(x) = (5-2x 3 ) 2 14) 15) To remodel a bathroom, a contractor charges $25 per hour plus material costs, which amount to $4000. Therefore, the total cost to remodel the bathroom is given by f(x) = 25x where x is the number of hours the contractor works. Find a formula for f -1 (x). What does f -1 (x) compute? 15) 16) If 5 x = 4,what does 5-3x equal? 16) 17) If 5 -x = 1 4, what does 25x equal? 17) 18) Define the number e. 18) 19) e 6x - 1 = (e 2 ) -x 19) 20) Suppose that f(x) = 5 x + 7. What is f(3)? What point is on the graph of f? 20) 21) 4 7-3x = ) 22) (e x ) x e 18 = e 9x 22) 2

3 Use a calculator to evaluate the expression. Round your answer to three decimal places e log 90 + ln 7 23) log 3 + ln 40 23) Find the domain of the function. 24) f(x) = log10 x + 6 x ) Change the logarithmic expression to an equivalent expression involving an exponent. 25) logb 49 = ) Find the domain of the function. 26) f(x) = log5(9 - x 2 ) 26) 27) log42 (x 2 - x) = 1 27) 28) Suppose that f(x) = 3 x + 2. If f(x) = 731, what is x? 28) Use the properties of logarithms to find the exact value of the expression. Do not use a calculator. 29) 10 log 30 - log 5 29) 30) ln x + 7 = 2 30) 31) e x + 2 = 4 31) 32) ph = -log10[h + ] Find the [H+] if the ph = ) Use the properties of logarithms to find the exact value of the expression. Do not use a calculator. 33) e log e ) Express y as a function of x. The constant C is a positive number. 34) ln y = ln 4x + ln C 34) 35) log11 x 2 = 4 35) Write as the sum and/or difference of logarithms. Express powers as factors. 36) log 4 5x 36) 3

4 Express as a single logarithm. 37) 2 log log 5 (x - 1) log 5 x 37) Use the Change-of-Base Formula and a calculator to evaluate the logarithm. Round your answer to three decimal places. 38) log ) Write as the sum and/or difference of logarithms. Express powers as factors. (x + 6)(x - 2) 3/4 39) ln, x > 2 39) (x - 8) 2 40) The formula D = 8e -0.04h can be used to find the number of milligrams D of a certain drug in a patient's bloodstream h hours after the drug has been given. When the number of milligrams reaches 3, the drug is to be given again. What is the time between injections? 40) Express as a single logarithm. 41) log a 3 + log a (x 3-4) + log a 2 41) 42) Find the value of log3 4 log4 5 log5 6 log6 7 log7 8 log ) Express y as a function of x. The constant C is a positive number. 43) 3ln y = 1 3 ln x - ln x2-1 + ln C 43) x4/3 Write as the sum and/or difference of logarithms. Express powers as factors. 44) log 5 x 4 y 8 44) 45) log 4 (x - 1) = 3 45) 46) log 6 (2x + 8) = log 6 (2x + 3) 46) 47) log 2 (x + 4) + log 2 (x - 2) = 4 47) 48) log (4 + x) - log (x - 5) = log 4 48) 49) log (3x) = log 2 + log (x - 1) 49) 50) log2(3x - 2) - log2(x - 5) = 4 50) 4

5 Express irrational answers in exact form and as a decimal rounded to 3 decimal places. 51) ln x + ln (x + 5) = -4 51) 52) f(x) = log3(x + 1) and g(x) = log3(x - 4). Solve g(x) = 129. What point is on the graph of g? 52) 53) The ph of a solution ranges from 0 to 14. An acid has a ph less than 7. Pure water is neutral and has a ph of 7. The ph of a solution is given by ph = - log x where x represents the concentration of the hydrogen ions in the solution in moles per liter. Find the hydrogen ion concentration if the ph = ) 54) f(x) = log4(x + 2) and g(x) = log4(2x - 1). Solve f(x) = g(x). 54) 55) The function f(x) = ln (x + 1) models the average number of free-throws a basketball player can make consecutively during practice as a function of time, where x is the number of consecutive days the basketball player has practiced for two hours. After how many days of practice can the basketball player make an average of 7 consecutive free throws? 55) 56) 3 (1 + 2x) = ) Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 57) 6 3x = ) 58) e 5x = 2 58) 59) 3 5 2t - 1 = 75 59) Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 60) e x + 8 = 4 60) 61) 9 7x + 3 = 27 61) 62) If f(x) = 2 x + 3 and g(x) = 2 -x + 5, find the point of intersection of the graphs of f and g by solving f(x) = g(x). 62) 63) The formula A = 225e 0.048t models the population of a particular city, in thousands, t years after When will the population of the city reach 300 thousand? 63) 5

6 64) The first recorded population of a particular country was 22 million, and the population was recorded as 34 million 10 years later. The exponential growth function A =22e kt describes the population of this country t years since the first recording. Use the fact that 10 years later the population increased by 12 million to find k to three decimal places. 64) Solve the given logarithmic equation. 65) 2 + log3(2x + 5) - log3 x = 4 65) Graph the function using a graphing utility and the Change-of-Base Formula. 66) y = log3(x - 3) Think on what transformation took place. Does it agree with the graph? Find x- and y-intercepts. Write the equation of the asymptote. 66) Use the graph of the given one-to-one function to sketch the graph of the inverse function. For convenience, the graph of y = x is also given. 67) 67) 68) The size P of a small herbivore population at time t (in years) obeys the function P(t) = 500e 0.17t if they have enough food and the predator population stays constant. After how many years will the population reach 1500? 68) 69) A culture of bacteria obeys the law of uninhibited growth. If 140,000 bacteria are present initially and there are 609,000 after 6 hours, how long will it take for the population to reach one million? 69) 6

7 70) The half-life of silicon-32 is 710 years. If 40 grams is present now, how much will be present in 200 years? (Round your answer to three decimal places.) 70) 71) A fossilized leaf contains 14% of its normal amount of carbon 14. How old is the fossil (to the nearest year)? Use 5600 years as the half-life of carbon ) 72) Strontium 90 decays at a constant rate of 2.44% per year. Therefore, the equation for the amount P of strontium 90 after t years is P = P0 e t. How long will it take for 15 grams of strontium to decay to 5 grams? Round answer to 2 decimal places. 72) 73) The amount of a certain drug in the bloodstream is modeled by the function y = y0 e t, where y0 is the amount of the drug injected (in milligrams) and t is the elapsed time (in hours). Suppose that 10 milligrams are injected at 10:00 A.M. If a second injection is to be administered when there is 1 milligram of the drug present in the bloodstream, approximately when should the next dose be given? Express your answer to the nearest quarter hour. 74) A thermometer reading 79 F is placed inside a cold storage room with a constant temperature of 35 F. If the thermometer reads 74 F in 13 minutes, how long before it reaches 57 F? Assume the cooling follows Newton's Law of Cooling: U = T + (Uo - T)e kt. (Round your answer to the nearest whole minute.) 75) A cup of coffee is heated to 194 and is then allowed to cool in a room whose air temperature is 72. After 11 minutes, the temperature of the cup of coffee is 140. Find the time needed for the coffee to cool to a temperature of ) The logistic growth model P(t) = represents the population of a e-0.325t bacterium in a culture tube after t hours. What was the initial amount of bacteria in the population? ) The logistic growth model P(t) = represents the population of a species e-0.188t introduced into a new territory after t years. When will the population be 80? 78) A life insurance company uses the following rate table for annual premiums for women for term life insurance. Use a graphing utility to fit an exponential function to the data. Predict the annual premium for a woman aged 70 years. Age Premium $103 $133 $190 $255 $360 $503 $818 73) 74) 75) 76) 77) 78) 79) After introducing an inhibitor into a culture of luminescent bacteria, a scientist monitors the luminosity produced by the culture. Use a graphing utility to fit a logarithmic function to the data. Predict the luminosity after 20 hours. Time, hrs Luminosity ) 7

8 80) A mechanic is testing the cooling system of a boat engine. He measures the engine's temperature over time. Use a graphing utility to fit a logistic function to the data. What is the carrying capacity of the cooling system? time, min temperature, F ) 8

9 Answer Key Testname: REVCHAP5-NO-MC 1) 25 2) f(x) = 1 x, g(x) = x 2-9 3) 18x x 4) {x x -2, x -4} 5) S(t) = 1 18 t4 + 4 t 2 6) {x x -8} 7) f-1(x) = 8) C = x p 40 9) f -1 (x) = 6x + 6-9x ) (a) V(W) = ( W ) (b) x 11) 4-15x 12) {x x 5, x 54} 13) 2 2x ) f(x) = x 2 ; g(x) = 5-2x 3 15) f -1 (x) = x - 160; This computes the number of hours worked if the total cost is x dollars ) ) 16 18) All of these. 1 19) 8 20) 132; (3, 132) 21) {3} 22) {6, 3} 23) ) (-, -6) (4, ) 25) b 2/3 = 49 26) (-3, 3) 27) {-6, 7} 28) 6 29) 6 30) {e 4-7} 31) {ln 4-2} 32) ) 9 34) y = 4Cx 9

10 Answer Key Testname: REVCHAP5-NO-MC 35) {121, -121} 36) 1 2 log log 4 x 37) log5 9 4 x - 1 x 38) ) 3 4 ln (x + 6) ln (x - 2) - 3 ln (x - 8) 2 40) hr 41) log a ( 6x 3-24) 42) 2 3 Cx 5/3 43) y = x ) 4 log 5 x - 8 log 5 y 45) {65} 46) 47) {4} 48) {8} 49) ) {6} 51) e ) {5}, (5, 129) 53) ) {3}, (3, log4(5)) 55) 100 days 56) {2} 57) {0.25} 58) {0.14} 3 59) 2 60) {-6.61} 61) ) (1, 16) 63) ) ) x =

11 Answer Key Testname: REVCHAP5-NO-MC 66) 67) 68) 6.46 yrs 69) hours 70) ) 15,856 72) years. 73) 3:45 P.M 74) 75 minutes 75) 26.4 minutes 76) 30 77) years 78) y = 8.94e 0.068x, $ ) y = ln (x), ) y =, 315 F e-0.246x 11