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All Textbook Solutions for Finite Mathematics for Business, Economics, Life Sciences and Social Sciences
For each rational function in Problems 23-28, (A) Find the intercepts for the graph. (B) Determine the domain. (C) Find any vertical or horizontal asymptotes for the graph. (D) Sketch any asymptotes as dashed lines. Then sketch a graph of y=fx fx=42xx4For each rational function in Problems 23-28, (A) Find the intercepts for the graph. (B) Determine the domain. (C) Find any vertical or horizontal asymptotes for the graph. (D) Sketch any asymptotes as dashed lines. Then sketch a graph of y=fx fx=33xx2Compare the graph of y=2x4 to the graph of y=2x45x2+x+2 in the following two viewing windows: A5x5,5y5B5x5,500y500Compare the graph of y=x3 to the graph of y=x32x+2 in the following two viewing windows: A5x5,5y5B5x5,500y500Compare the graph of y=x5 to the graph of y=x5+4x34x+1 in the following two viewing windows: A5x5,5y5B5x5,500y500Compare the graph of y=x5 to the graph of y=x5+5x35x+2 in the following two viewing windows: A5x5,5y5B5x5,500y500In Problems 33-40, find the equation of any horizontal asymptote. fx=5x3+2x36x37x+1In Problems 33-40, find the equation of any horizontal asymptote. fx=6x4x3+24x4+10x+5In Problems 33-40, find the equation of any horizontal asymptote. fx=15x+x22+3x+4x2In Problems 33-40, find the equation of any horizontal asymptote. fx=8x31+2x3In Problems 33-40, find the equation of any horizontal asymptote. fx=x4+2x2+11x5In Problems 33-40, find the equation of any horizontal asymptote. fx=3+5xx2+x+3In Problems 33-40, find the equation of any horizontal asymptote. fx=x2+6x+1x5In Problems 33-40, find the equation of any horizontal asymptote. fx=x2+x4+1x3+2x4In Problems 41-46, find the equations of any vertical asymptotes. fx=x2+1x21x29In Problems 41-46, find the equations of any vertical asymptotes. fx=2x+5x24x216In Problems 41-46, find the equations of any vertical asymptotes. fx=x2x6x23x10In Problems 41-46, find the equations of any vertical asymptotes. fx=x28x+7x2+7x8In Problems 41-46, find the equations of any vertical asymptotes. fx=x2+3xx336xIn Problems 41-46, find the equations of any vertical asymptotes. fx=x2+x2x33x2+2xFor each rational function in Problems 47-52, (A) Find any intercepts for the graph. (B) Find any vertical and horizontal asymptotes for the graph. (C) Sketch any asymptotes as dashed lines. Then sketch a graph of f. (D Graph the function in a standard viewing window using a graphing calculator. fx=2x2x2x6For each rational function in Problems 47-52, (A) Find any intercepts for the graph. (B) Find any vertical and horizontal asymptotes for the graph. (C) Sketch any asymptotes as dashed lines. Then sketch a graph of f. (D Graph the function in a standard viewing window using a graphing calculator. fx=3x2x2+x6For each rational function in Problems 47-52, (A) Find any intercepts for the graph. (B) Find any vertical and horizontal asymptotes for the graph. (C) Sketch any asymptotes as dashed lines. Then sketch a graph of f. (D Graph the function in a standard viewing window using a graphing calculator. fx=62x2x29For each rational function in Problems 47-52, (A) Find any intercepts for the graph. (B) Find any vertical and horizontal asymptotes for the graph. (C) Sketch any asymptotes as dashed lines. Then sketch a graph of f. (D Graph the function in a standard viewing window using a graphing calculator. fx=33x2x24For each rational function in Problems 47-52, (A) Find any intercepts for the graph. (B) Find any vertical and horizontal asymptotes for the graph. (C) Sketch any asymptotes as dashed lines. Then sketch a graph of f. (D Graph the function in a standard viewing window using a graphing calculator. fx=4x+24x2+x6For each rational function in Problems 47-52, (A) Find any intercepts for the graph. (B) Find any vertical and horizontal asymptotes for the graph. (C) Sketch any asymptotes as dashed lines. Then sketch a graph of f. (D Graph the function in a standard viewing window using a graphing calculator. fx=5x10x2+x12Write an equation for the lowest-degree polynomial function with the graph and intercepts shown in the figure.Write an equation for the lowest-degree polynomial function with the graph and intercepts shown in the figure.Write an equation for the lowest-degree polynomial function with the graph and intercepts shown in the figure.Write an equation for the lowest-degree polynomial function with the graph and intercepts shown in the figure.Average cost. A company manufacturing snowboards has fixed costs of $200 per day and total costs of $3,800 per day at a daily output of 20 boards. (A) Assuming that the total cost per day, Cx, is linearly related to the total output per day, x, write an equation for the cost function. (B) The average cost per board for an output of x boards is given by Cx=Cx/x. Find the average cost function. (C) Sketch a graph of the average cost function, including any asymptotes, for 1x30 (D) What does the average cost per board tend to as production increases?Average cost. A company manufacturing snowboards has fixed costs of $300 per day and total costs of $5,100 per day at a daily output of 20 boards. (A) Assuming that the total cost per day, Cx, is linearly related to the total output per day, x, write an equation for the cost function. (B) The average cost per board for an output of x boards is given by Cx=Cx/x. Find the average cost function. (C) Sketch a graph of the average cost function, including any asymptotes, for 1x30. (D) What does the average cost per board tend to as production increases?Replacement time. An office copier has an initial price of $2,500. A service contract costs $200 for the first year and increases $50 per year thereafter. It can be shown that the total cost of the copier after n years is given by Cn=2,500+175n+25n2 The average cost per year for n years is given by Cn=Cn/n. (A) Find the rational function C. (B) Sketch a graph of Cfor2n20. (C) When the average cost per year at a minimum, and what is is the minimum average annual cost? [Hint: Refer to the sketch in part (B) and evaluate Cn at appropriate integer values until a minimum value is found.] The time when the average cost is minimum is frequently referred to as the replacement time for the piece of equipment. (D) Graph the average cost function C on a graphing calculator and use an appropriate command to find when the average annual cost is at a minimum.Minimum average cost. Financial analysts in a company that manufactures DVD players arrived at the following daily cost equation for manufacturing x DVD players per day: Cx=x2+2x+2,000 The average cost per unit at a production level of x players per day is Cx=Cx/x. (A) Find the rational function C. (B) Sketch a graph of Cfor5x150. (C) For what daily production level (to the nearest integer) is the average cost per unit at a minimum, and what is the minimum average cost per player (to the nearest cent)? [Hint: Refer to the sketch in part (B) and evaluate Cx at appropriate integer values until a minimum value is found.] (D) Graph the average cost function C on a graphing calculator and use an appropriate command to find the daily production level (to the nearest integer) at which the average cost per player is at a minimum. What is the minimum average cost to the nearest cent?Minimum average cost. A consulting firm, using statistical methods, provided a veterinary clinic with the cost equation Cx=0.00048x5003+60,000100x1,000 where Cx is the cost in dollars for handling x cases per month. The average cost per case is given by Cx=Cx/x (A) Write the equation for the average cost function C. (B) Graph C on a graphing calculator. (C) Use an appropriate command to find the monthly caseload for the minimum average cost per case. What is the minimum average cost per case?Minimum average cost. The financial department of a hospital, using statistical methods, arrived at a cost equation Cx=20x3360x2+2,300x1,0001x12 where Cx is the cost in thousands of dollars for handling x thousand cases per month. The average cost per case is given by Cx=Cx/x (a) Write the equation for the average cost function C. (b) Graph C on a graphing calculator. (c) Use an appropriate command to find the monthly caseload for the minimum average cost per case. What is the minimum average cost per case to the nearest dollar?Diet. Table 3 shows the per capita consumption of ice cream in the United States for selected years since 1987. (A) Let x represent the number of years since 1980 and find a cubic regression polynomial for the per capita consumption of ice cream. (B) Use the polynomial model from part (A) to estimate (to the nearest tenth of a pound) the per capita consumption of ice cream in 2023.Diet. Refer to Table 4. (A) Let x represent the number of years since 2000 and find a cubic regression polynomial for the per capita consumption of eggs. (B) Use the polynomial model from part (A) to estimate (to the nearest integer) the per capita consumption of eggs in 2023.Physiology. In a study on the speed of muscle contraction in frogs under various loads, researchers W. O. Ferns and J. Marsh found that the speed of contraction decreases with increasing loads. In particular, they found that the relationship between speed of contraction v (in centimeters per second) and load x (in grams) is given approximately by vx=26+0.06xxx5 (A) What does vx approach as x increases? (B) Sketch a graph of function v.Learning theory. In 1917, L. L. Thurstone, a pioneer in quantitative learning theory, proposed the rational function fx=ax+cx+c+b to model the number of successful acts per unit time that a person could accomplish after x practice sessions. Suppose that for a particular person enrolled in a typing class, fx=55x+1x+8x0 where fx is the number of words per minute the person is able to type after x weeks of lessons. (A) What does fx approach as x increases? (B) Sketch a graph of function f, including any vertical or horizontal asymptotes.Marriage. Table 5 shows the marriage and divorce rates per 1,000 population for selected years since 1960. (A) Let x represent the number of years since 1960 and find a cubic regression polynomial for the marriage rate. (B) Use the polynomial model from part (A) to estimate the marriage rate (to one decimal place) for 2025.Divorce. Refer to Table 5. (A) Let x represent the number of years since 1960 and find a cubic regression polynomial for the divorce rate. (B) Use the polynomial model from part (A) to estimate the divorce rate (to one decimal place) for 2025.Graph the functions fx=ex,gx=2x,andhx=3x on the same set of coordinate axes. At which values of x do the graphs intersect? For positive values of x, which of the three graphs lies above the other two? Below the other two? How does your answer change for negative values of x ?Suppose that $1,000 is deposited in a savings account at an annual rate of 5. Guess the amount in the account at the end of 1 year if interest is compounded 1 quarterly, 2 monthly, 3 daily, 4 hourly. Use the compound interest formula to compute the amounts at the end of 1 year to the nearest cent. Discuss the accuracy of your initial guesses.Sketch a graph of y=124x,2x2.Refer to the exponential growth model for cholera in Example 2. If we start with 55 bacteria, how many bacteria (to the nearest unit) will be present (A) In 0.85 hour? (B) In 7.25 hours?Refer to the exponential decay model in Example 3. How many milligrams of 14C would have to be present at the beginning in order to have 25 milligrams present after 18,000 years? Compute the answer to the nearest milligram.Table 3 gives the market value of a midsize sedan (in dollars) x years after its purchase. Find an exponential regression model of the form y=abx for this data set. Estimate the purchase price of the sedan. Estimate the value of the sedan 10 years after its purchase. Round answers to the nearest dollar.If you deposit $5,000 in an account paying 9 compounded daily, how much will you have in the account in 5 years? Compute the answer to the nearest cent.If you deposit $5,000 in an account paying 9 compounded continuously, how much will you have in the account in 5 years? Compute the answer to the nearest cent.Match each equation with the graph of f,g,h,ork in the figure. (A) y=2x (B) y=0.2x (C) y=4x (D) y=13xMatch each equation with the graph of f,g,h,ork in the figure. (A) y=14x (B) y=0.5x (C) y=5x (D) y=3xGraph each function in Problems 3-10 over the indicated interval. y=5x;2,2Graph each function in Problems 3-10 over the indicated interval. y=3x;3,3Graph each function in Problems 3-10 over the indicated interval. y=15x=5x;2,2Graph each function in Problems 3-10 over the indicated interval. y=13x=3x;3,3Graph each function in Problems 3-10 over the indicated interval. fx=5x;2,2Graph each function in Problems 3-10 over the indicated interval. gx=3x;3,3Graph each function in Problems 3-10 over the indicated interval. y=ex;3,3Graph each function in Problems 3-10 over the indicated interval. y=ex;3,3In Problems11-18, describe verbally the transformations that can be used to obtain the graph of g from the graph of f (see Section2.2 ). gx=2x;fx=2xIn Problems11-18, describe verbally the transformations that can be used to obtain the graph of g from the graph of f (see Section2.2 ). gx=2x2;fx=2xIn Problems11-18, describe verbally the transformations that can be used to obtain the graph of g from the graph of f (see Section2.2 ). gx=3x+1;fx=3xIn Problems11-18, describe verbally the transformations that can be used to obtain the graph of g from the graph of f (see Section2.2 ). gx=3x;fx=3xIn Problems11-18, describe verbally the transformations that can be used to obtain the graph of g from the graph of f (see Section2.2 ). gx=ex+1;fx=exIn Problems 11-18, describe verbally the transformations that can be used to obtain the graph of g from the graph of f (see Section 2.2 ). gx=ex2;fx=exIn Problems 11-18, describe verbally the transformations that can be used to obtain the graph of g from the graph of f (see Section2.2 ). gx=2ex+2;fx=exIn Problems 11-18, describe verbally the transformations that can be used to obtain the graph of g from the graph of f (see Section2.2 ). gx=0.5ex1;fx=exUse the graph of f shown in the figure to sketch the graph of each of the following. Ay=fx1By=fx+2Cy=3fx2Dy=2fx3Use the graph of f shown in the figure to sketch the graph of each of the following. Ay=fx+2By=fx3Cy=2fx4Dy=4fx+2In Problems 21-26, graph each function over the indicated interval. ft=2t/10;30,30In Problems 21-26, graph each function over the indicated interval. Gt=3t/100;200,200In Problems 21-26, graph each function over the indicated interval. y=3+e1+x;4,2In Problems 21-26, graph each function over the indicated interval. y=2+ex2;1,5In Problems 21-26, graph each function over the indicated interval. y=ex;3,3In Problems 21-26, graph each function over the indicated interval. y=ex;3,3Find all real numbers a such that a2=a2. Explain why this does not violate the second exponential function property in Theorem 2 on page 98.Find real numbers aandb such that ab but a4=b4. Explain why this does not violate the third exponential function property in Theorem 2 on page 98.In Problems 29-38, solve each equation for x. 22x+5=210130EIn Problems 29-38, solve each equation for x. 7x2 =73x+10In Problems 29-38, solve each equation for x. 5x2x=542In Problems 29-38, solve each equation for x. 3x+95=32x5In Problems 29-38, solve each equation for x. 3x+43=523In Problems 29-38, solve each equation for x. x+52=2x14236E37EIn Problems 29-38, solve each equation for x. 4x+14=5x104In Problems 39-46, solve each equation for x. (Remember: ex0andex0 for all values of x ). xex+7ex=0In Problems 39-46, solve each equation for x. (Remember: ex0andex0 for all values of x ). 10xex5ex=0In Problems 39-46, solve each equation for x. (Remember: ex0andex0 for all values of x ). 2x2ex8ex=0In Problems 39-46, solve each equation for x. (Remember: ex0andex0 for all values of x ). x2ex9ex=0In Problems 39-46, solve each equation for x. (Remember: ex0andex0 for all values of x ). e4xe=0In Problems 39-46, solve each equation for x. (Remember: ex0andex0 for all values of x ). e4x+e=0In Problems 39-46, solve each equation for x. (Remember: ex0andex0 for all values of x ). e3x1+e=0In Problems 39-46, solve each equation for x. (Remember: ex0andex0 for all values of x ). e3x1e=0Graph each function in Problems 47-50 over the indicated interval. hx=x2x;5,0Graph each function in Problems 47-50 over the indicated interval. mx=x3x;0,3Graph each function in Problems 47-50 over the indicated interval. N=1001+et;0,5Graph each function in Problems 47-50 over the indicated interval. N=2001+3et;0,5Continuous compound interest. Find the value of an investment of $10,000 in 12 years if it earns an annual rate of 3.95 compounded continuously.Continuous compound interest. Find the value of an investment of $24,000 in 7 years if it earns an annual rate of 4.35 compounded continuously.Compound growth. Suppose that $2,500 is invested at 7 compounded quarterly. How much money will be in the account in (A) 34 year? (B) 15 years? Compute answers to the nearest cent.Compound growth. Suppose that $4,000 is invested at 6 compounded weekly. How much money will be in the account in (A) 12 year? (B) 10 years? Compute answers to the nearest cent.Finance. A person wishes to have $15,000 cash for a new car 5 years from now. How much should be placed in an account now, if the account pays 6.75 compounded weekly? Compute the answer to the nearest dollar.Finance. A couple just had a baby. How much should they invest now at 5.5 compounded daily in order to have 40,000 for the child’s education 17 years from now? Compute the answer to the nearest dollar.Money growth. BanxQuote operates a network of websites providing real-time market data from leading financial providers. The following rates for 12-month certificates of deposit were taken from the websites: (A) Stonebridge Bank, 0.95 compounded monthly (B) DeepGreen Bank, 0.80 compounded daily (C) Provident Bank, 0.85 compounded quarterly Compute the value of $10,000 invested in each account at the end of 1 year.Money growth. Refer to Problem 57. The following rates for 60-month certificates of deposit were also taken from BanxQuote websites: (A) Oriental Bank compounded quarterly (B) BMW Bank of North America, 1.30 compounded monthly (C) BankFirst Corporation, 1.25 compounded daily. Compute the value of $10,000 invested in each account at the end of 5 years.Advertising. A company is trying to introduce a new product to as many people as possible through television advertising in a large metropolitan area with 2 million possible viewers. A model for the number of people N (in millions) who are aware of the product after t days of advertising was found to be N=21e0.037t Graph this function for 0t50. What value does N approach as t increases without bound?Learning curve. People assigned to assemble circuit boards for a computer manufacturing company undergo on-the-job training. From past experience, the learning curve for the average employee is given by N=401e0.12t where N is the number of boards assembled per day after t days of training. Graph this function for 0t30. What is the maximum number of boards an average employee can be expected to produce in 1 day?Internet users. Table 4 shows the number of individuals worldwide who could access the internet from home for selected years since 2000. (A) Let x represent the number of years since 2000 and find an exponential regression model y=abx for the number of internet users. (B) Use the model to estimate the number of internet users in 2024.Mobile data traffic. Table 5 estimates of mobile data traffic, in exabytes 1018bytes per month, for years from 2015 to 2020. (A) Let x represent the number of years since 2015 and find an exponential regression model y=abx for the mobile data traffic. (B) Use the model to estimate the mobile data traffic in 2025.Marine biology. Marine life depends on the microscopic plant life that exists in the photic zone, a zone that goes to a depth where only 1 of surface light remains. In some waters with a great deal of sediment, the photic zone may go down only 15 to 20 feet. In some murky harbors, the intensity of light d feet below the surface is given approximately by I=I0e0.23d where I0 is the intensity of light at the surface. What percentage of the surface light will reach a depth of (A) 10 feet? (B) 20 feet?Marine biology. Refer to Problem 63. Light intensity I relative to depth d (in feet) for one of the clearest bodies of water in the world, the Sargasso Sea, can be approximated by I=I0e0.00942d where I0 is the intensity of light at the surface. What percentage of the surface light will reach a depth of (A) 50 feet? (B) 100 feet?Population growth. In 2015, the estimated population of South Sudan was 12 million with a relative growth rate of 4.02. (A) Write an equation that models the population growth in South Sudan, letting 2015 be year 0. (B) Based on the model, what is the expected population of South Sudan in 2025 ?Population growth. In 2015, the estimated population of Brazil was 204 million with a relative growth rate of 0.77. (A) Write an equation that model the population growth in Brazil, letting 2015 be year 0. (B) Based on the model, what is the expected population of Brazil in 2030 ?Population growth. In 2015, the estimated population of Japan was 127 million with a relative growth rate of 0.16. (A) Write an equation that models the population growth in Japan, letting 2015 be year 0. (B) Based on the model, what is the expected population in Japan in 2030 ?World population growth. From the dawn of humanity to 1830, world population grew to one billion people. In 100 more years (by 1930 ) it grew to two billion, and 3 billion more were added in only 60 years (by 1990 ). In 2016, the estimated world population was 7.4 billion with a relative growth rate of 1.13. (A) Write an equation that models the world population growth, letting 2016 be year 0. (B) Based on the model, what is the expected world population (to the nearest hundred million) in 2025 ? In 2033 ?Graph fx=2xandgx=x2. For a range value of 4, what are the corresponding domain values for each function? Which of the two functions is one-to-one? Explain why.Discuss how you could find y=log538.25 using either natural or common logarithms on a calculator. [Hint: Start by rewriting the equation in exponential form.]Change each logarithmic form to an equivalent exponential form: Alog39=2Blog42=12Clog319=2Change each exponential form to an equivalent logarithmic form: A49=72B3=9C13=31Find y,b,orx, as indicated. AFindy:y=log927BFindx:log3x=1CFindb:logb1,000=3Write in simpler form, as in Example 4. AlogbRSTBlogbRS2/3C2ulog2bDlog2xlog2bFind x so that 3logb2+12logb25logb20=logbx.Solve: log3x+log3x3=log310Use a calculator to evaluate each to six decimal places: Alog0.013529Bln28.69328Cln0.438Find x to four decimal places, given the indicated logarithm: Alnx=5.062Blogx=2.0821Solve for x to four decimal places: A10x=7Bex=6C4x=5How long (to the next whole year) will it take money to triple if it is invested at 13 compounded annually?Refer to Example 11. Use the model to predict the home ownership rate in the United States in 2030 (to the nearest tenth of a percent).For Problems 1-6, rewrite in equivalent exponential form. log327=3For Problems 1-6, rewrite in equivalent exponential form. log232=5For Problems 1-6, rewrite in equivalent exponential form. log101=0For Problems 1-6, rewrite in equivalent exponential form. loge1=0For Problems 1-6, rewrite in equivalent exponential form. log48=32For Problems 1-6, rewrite in equivalent exponential form. log927=32For Problems 7-12, rewrite in equivalent logarithmic form. 49=72For Problems 7-12, rewrite in equivalent logarithmic form. 36=62For Problems 7-12, rewrite in equivalent logarithmic form. 8=43/2For Problems 7-12, rewrite in equivalent logarithmic form. 9=272/3For Problems 7-12, rewrite in equivalent logarithmic form. A=buFor Problems 7-12, rewrite in equivalent logarithmic form. M=bxIn Problems 13-22, evaluate the expression without using a calculator. log101,000,000In Problems 13-22, evaluate the expression without using a calculator. log1011,000In Problems 13-22, evaluate the expression without using a calculator. log101100,000In Problems 13-22, evaluate the expression without using a calculator. log1010,000In Problems 13-22, evaluate the expression without using a calculator. log2128In Problems 13-22, evaluate the expression without using a calculator. log2164In Problems 13-22, evaluate the expression without using a calculator. lne320EIn Problems 13-22, evaluate the expression without using a calculator. eln3In Problems 13-22, evaluate the expression without using a calculator. lne1For Problems 23-28, write in simpler form, as in Example 4. logbPQFor Problems 23-28, write in simpler form, as in Example 4. logbFGFor Problems 23-28, write in simpler form, as in Example 4. logbL5For Problems 23-28, write in simpler form, as in Example 4. logbw15For Problems 23-28, write in simpler form, as in Example 4. 3plog3qFor Problems 23-28, write in simpler form, as in Example 4. log3Plog3RFor Problems 29-38, find x,y, or b without using a calculator. log10x=1.For Problems 29-38, find x,y, or b without using a calculator. log10x=1For Problems 29-38, find x,y, or b without using a calculator. logb64=3For Problems 29-38, find x,y, or b without using a calculator. logb125=2For Problems 29-38, find x,y, or b without using a calculator. log218=yFor Problems 29-38, find x,y, or b without using a calculator. log497=yFor Problems 29-38, find x,y, or b without using a calculator. logb81=4For Problems 29-38, find x,y, or b without using a calculator. logb10,000=2For Problems 29-38, find x,y, or b without using a calculator. log4x=32For Problems 29-38, find x,y, or b without using a calculator. log8x=53In Problems 39-46, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter example. Every polynomial function is one-to-one.In Problems 39-46, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter example. Every polynomial function of odd degree is one-to-one.In Problems 39-46, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter example. If g is the inverse of a function f, then g is one-to-one.In Problems 39-46, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter example. The graph of a one-to-one function intersects each vertical line exactly once.In Problems 39-46, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter example. The inverse of fx=2x isgx=x/2.In Problems 39-46, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter example. The inverse of fx=x2isgx=x.In Problems 39-46, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter example. If f is one-to-one, then the domain of f is equal to the range of f.In Problems 39-46, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter example. If g is the inverse of a function f, then f is the inverse of g.Find x in Problems 47-54. logbx=23logb8+12logb9logb6Find x in Problems 47-54. logbx=23logb27+2logb2logb3Find x in Problems 47-54. logbx=32logb423logb8+2logb2Find x in Problems 47-54. logbx=3logb2+12logb25logb20Find x in Problems 47-54. logbx+logbx4=logb21Find x in Problems 47-54. logbx+2+logbx=logb24Find x in Problems 47-54. log10x1log10x+1=1Find x in Problems 47-54. log10x+6log10x3=1Graph Problems 55 and 56 by converting to exponential form first. y=log2x2Graph Problems 55 and 56 by converting to exponential form first. y=log3x+2Explain how the graph of the equation in Problem 55 can be obtained from the graph of y=log2x using a simple transformation (see Section 2.2 ).Explain how the graph of the equation in Problem 56 can be obtained from the graph of y=log3x using a simple transformation (see Section 2.2).What are the domain and range of the function defined by y=1+lnx+1 ?What are the domain and range of the function defined by y=logx11 ?For Problems 61 and 62, evaluate to five decimal places using a calculator. Alog3,527.2Blog0.0069132Cln277.63Dln0.040883For Problems 61 and62, evaluate to five decimal places using a calculator. Alog72.604Blog0.033041Cln40,257Dln0.0059263For Problems 63 and 64, find x to four decimal places. Alogx=1.1285Blogx=2.0497Clnx=2.7763Dlnx=1.8879For Problems 63 and 64, find x to four decimal places. Alogx=2.0832Blogx=1.1577Clnx=3.1336Dlnx=4.3281For Problems 65-70, solve each equation to four decimal places. 10x=12For Problems 65-70, solve each equation to four decimal places. 10x=153For Problems 65-70, solve each equation to four decimal places. ex=4.304For Problems 65-70, solve each equation to four decimal places. ex=0.3029For Problems 65-70, solve each equation to four decimal places. 1.00512t=3For Problems 65-70, solve each equation to four decimal places. 1.024t=2Graph Problems 71-78 using a calculator and point-by-point plotting. Indicate increasing and decreasing intervals. y=lnxGraph Problems 71-78 using a calculator and point-by-point plotting. Indicate increasing and decreasing intervals. y=lnxGraph Problems 71-78 using a calculator and point-by-point plotting. Indicate increasing and decreasing intervals. y=lnxGraph Problems 71-78 using a calculator and point-by-point plotting. Indicate increasing and decreasing intervals. y=lnxGraph Problems 71-78 using a calculator and point-by-point plotting. Indicate increasing and decreasing intervals. y=2lnx+2Graph Problems 71-78 using a calculator and point-by-point plotting. Indicate increasing and decreasing intervals. y=2lnx+2Graph Problems 71-78 using a calculator and point-by-point plotting. Indicate increasing and decreasing intervals. y=4lnx3Graph Problems 71-78 using a calculator and point-by-point plotting. Indicate increasing and decreasing intervals. y=4lnx3Explain why the logarithm of 1 for any permissible base is 0.Explain why 1 is not a suitable logarithmic base.Let px=lnx,qx=x,andrx=x. Use a graphing calculator to draw graphs of all three functions in the same viewing window for 1x16. Discuss what it means for one function to be larger than another on an interval, and then order the three functions from largest to smallest for 1x16Let p(x)=logx,qx=x3,andrx=x. Use a graphing calculator to draw graphs of all three functions in the same viewing window for 1x16. Discuss what it means for one function to be smaller than another on an interval, and then order the three functions from smallest to largest for 1x16.Doubling time. In its first 10 years the Gabelli Growth Fund produced an average annual return of 21.36. Assume that money invested in this fund continues to earn 21.36 compounded annually. How long (to the nearest year) will it take money invested in this fund to double?Doubling time. In its first 10 years the Janus Flexible Income Fund produced an average annual return of 9.58. Assume that money invested in this fund continues to earn 9.58 compounded annually. How long (to the nearest year) will it take money invested in this fund to double?Investing. How many years (to two decimal places) will it take $1,000 to grow to $1,800 if it is invested at 6 compounded quarterly? Compounded daily?Investing. How many years (to two decimal places) will it take $5,000 to grow to $7,500 if it is invested at 8 compounded semiannually? Compounded monthly?Continuous compound interest. How many years (to two decimal places) will it take an investment of $35,000 to grow to $50,000 if it is invested at 4.75 compounded continuously?Continuous compound interest. How many years (to two decimal places) will it take an investment of $17,000 to grow to $41,000 if it is invested at 2.95 compounded continuously?Supply and demand. A cordless screwdriver is sold through a national chain of discount stores. A marketing company established price-demand and price-supply tables (Tables 2 and 3 ), where x is the number of screwdrivers people are willing to buy and the store is willing to sell each month at a price of p dollars per screwdriver. (A) Find a logarithmic regression model y=a+blnx for the data in Table 2. Estimate the demand (to the nearest unit) at a price level of 50. (B) Find a logarithmic regression model y=a+blnx for the data in Table 3. Estimate the supply (to the nearest unit) at a price level of $50. (C) Does a price level of $50 represent a stable condition, or is the price likely to increase or decrease? Explain.Equilibrium point. Use the models constructed in Problem 89 to find the equilibrium point. Write the equilibrium price to the nearest cent and the equilibrium quantity to the nearest unit.Sound intensity: decibels. Because of the extraordinary range of sensitivity of the human ear (a range of over 1,000 million millions to 1 ), it is helpful to use a logarithmic scale, rather than an absolute scale, to measure sound intensity over this range. The unit of measure is called the decibel, after the inventor of the telephone, Alexander Graham Bell. If we let N be the number of decibels, I the power of the sound in question (in watts per square centimeter), and I0 the power of sound just below the threshold of hearing (approximately 1016 watt per square centimeter), then I=I010N/10 Show that this formula can be written in the form N=10logII0Sound intensity: decibels. Use the formula in Problem 91 withI0=1016W/cm2 to find the decibel ratings of the following sounds: (A) Whisper: 1013W/cm2 (B) Normal conversation: 3.161010W/cm2 (C) Heavy traffic: 108W/cm2 (D) Jet plane with afterburner: 101W/cm2Agriculture. Table 4 shows the yield (in bushels per acre) and the total production (in millions of bushels) for com in the United States for selected years since 1950. Let x represent years since 1900. Find a logarithmic regression model y=a+blnx for the yield. Estimate (to the nearest bushel per acre) the yield in 2024.Agriculture. Refer to Table 4. Find a logarithmic regression model y=a+blnx for the total production. Estimate (to the nearest million) the production in d 2024.World population. If the world population is now 7.4 billion people and if it continues to grow at an annual rate of 1.1 compounded continuously, how long (to the nearest year) would it take before there is only 1 square yard of land per person? (The Earth contains approximately 1.681014 square yards of land.)Archaeology: carbon -14 dating. The radioactive carbon -1414C in an organism at the time of its death decays according to the equation A=A0e0.000124t where t is time in years and A0 is the amount of 14C present at time t=0. (See Example 3 in Section 2.5.) Estimate the age of a skull uncovered in an archaeological site if 10 of the original amount of 14C is still present. [Hint: Find t such that A=0.1A0 ]In Problems 1-3, use point-by-point plotting to sketch the graph of each equation. y=5x2In Problems 1-3, use point-by-point plotting to sketch the graph of each equation. x2=y2In Problems 1-3, use point-by-point plotting to sketch the graph of each equation. y2=4x2Indicate whether each graph specifies a function:For fx=2x1 and gx=x22x, find: (a) f2+g1 (b) f0g4 (c) g2f3 (d) f3g2Write in logarithmic form using base e:u=eu.Write in logarithmic form using base 10:x=10y.Write in exponential form using base e:lnM=N.Write in exponential form using base 10:logu=v.Solve Problems for exactly without using a calculator.Solve Problems 10-12 for x exactly without using a calculator. logx36=2Solve Problems 10-12 for x exactly without using a calculator. log216=xSolve Problems 13-16 for x to three decimal places. 10x=143.7Solve Problems 13-16 for x to three decimal places. ex=503,000Solve Problems 13-16 for x to three decimal places. logx=3.105Solve Problems 13-16 for x to three decimal places. lnx=1.147Use the graph of function f in the figure to determine (to the nearest integer) xory as indicated. Ay=f0B4=fxCy=f3D3=fxEy=f6F1=fxSketch a graph of each of the functions in parts AD using the graph of function f in the figure below. Ay=fxBy=fx+4Cy=fx2Dy=fx+33Complete the square and find the standard form for the quadratic function fx=x2+4x Then write a brief verbal description of the relationship between the graph of f and the graph of y=x2.Match each equation with a graph of one of the functions f,g,m,orn in the figure Ay=x224By=x+22+4Cy=x22+4Dy=x+224Referring to the graph of function f in the figure for Problem 20 and using known properties of quadratic functions, find each of the following to the nearest integer: AInterceptsBVertexCMaximumorminimumDRangeIn Problems 22-25, each equation specifies a function. Determine whether the function is linear, quadratic, constant, or none of these. y=4x+3x2In Problems 22-25, each equation specifies a function. Determine whether the function is linear, quadratic, constant, or none of these. y=1+5x6In Problems 22-25, each equation specifies a function. Determine whether the function is linear, quadratic, constant, or none of these. y=74x2xIn Problems 22-25, each equation specifies a function. Determine whether the function is linear, quadratic, constant, or none of these. y=8x+2104xSolve Problems 26-33 for x exactly without using a calculator. logx+5=log2x3Solve Problems 26-33 for x exactly without using a calculator. 2lnx1=lnx25Solve Problems 26-33 for x exactly without using a calculator. 9x1=31+xSolve Problems 26-33 for x exactly without using a calculator. e2x=ex23Solve Problems 26-33 for x exactly without using a calculator. 2x2ex=3xexSolve Problems 26-33 for x exactly without using a calculator. log1/39=xSolve Problems 26-33 for x exactly without using a calculator. logx8=3Solve Problems 26-33 for x exactly without using a calculator. log9x=32Solve Problems 34-41 for x to four decimal places. x=3e1.49Solve Problems 34-41 for x to four decimal places. x=230100.161Solve Problems 34-41 for x to four decimal places. logx=2.0144Solve Problems 34-41 for x to four decimal places. lnx=0.3618Solve Problems 34-41 for x to four decimal places. 35=73xSolve Problems 34-41 for x to four decimal places. 0.01=e0.05xSolve Problems 34-41 for x to four decimal places. 8,000=4,0001.08xSolve Problems 34-41 for x to four decimal places. 52x3=7.08Find the domain of each function: Afx=2x5x2x6Bgx=3x5xFind the vertex form for fx=4x2+4x3 and then find the intercepts, the vertex, the maximum or minimum, and the range.Let fx=ex1andgx=lnx+2. Find all points of intersection for the graphs of f and g. Round answers to two decimal places.In Problems 45and46, use point-by-point plotting to sketch the graph of each function. fx=50x2+1In Problems 45and46, use point-by-point plotting to sketch the graph of each function. fx=662+x2If fx=5x+1, find and simplify each of the following in Problems 47-50. ff0If fx=5x+1, find and simplify each of the following in Problems 47-50. ff1If fx=5x+1, find and simplify each of the following in Problems 47-50. f2x1If fx=5x+1, find and simplify each of the following in Problems 47-50. f4xLet fx=32x. Find Af2Bf2+hCf2+hf2Df2+hf2h,h0Let fx=x23x+1. Find AfaBfa+hCfa+hfaDfa+hfah,h0Explain how the graph of mx=x4 is related to graph of y=x.Explain how the graph of gx=0.3x3+3 is related to the graph of y=x3.55REThe graph of a function f is formed by vertically stretching the graph of y=x by a factor of 2. and shifting it to the left 3 units and down 1 unit. Find an equation for function f and graph it for 5x5 and 5y5.In Problems 57-59, find the equation of any horizontal asymptote. fx=5x+4x23x+4In Problems 57-59, find the equation of any horizontal asymptote. fx=3x2+2x14x25x+3In Problems 57-59, find the equation of any horizontal asymptote. fx=x2+4100x+1In Problems 60 and 61, find the equations of any vertical asymptotes. fx=x2+100x2100In Problems 60 and 61, find the equations of any vertical asymptotes. fx=x2+3xx2+2xIn Problems 62-67, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter example. Every polynomial function is a rational function.In Problems 62-67, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter example. Every rational function is a polynomial function.In Problems 62-67, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter example. The graph of every rational function has at least one vertical asymptote.In Problems 62-67, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter example. The graph of every exponential function has a horizontal asymptote.In Problems 62-67, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter example. The graph of every logarithmic function has a vertical asymptote.In Problems 62-67, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter example. There exists a rational function that has both a vertical and horizontal asymptote.Sketch the graph of f for x0 fx=9+0.3xif0x205+0.2xifx20Sketch the graph of g for x0. fx=0.5x+5if0x101.2x2if10x302x26ifx30Write an equation for the graph shown in the form y=axh2+k, where a is either 1or1handk are integers.Given fx=0.4x2+3.2x+1.2 find the following algebraically (to one decimal place) without referring to a graph: AInterceptsBVertexCMaximumorminimumDRangeGiven fx=0.4x2+3.2x+1.2 in a graphing calculator and find the following (to one decimal place) using TRACE and appropriate commands: AInterceptsBVertexCMaximumorminimumDRangeNoting that =3.141592654... and 2=1.414213562... explain why the calculator results shown here are obvious. Discuss similar connections between the natural logarithmic function and the exponential function with base e.