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All Textbook Solutions for Precalculus

Determine whether the functions are inverses. a.fx=x+62andgx=2x6b.mx=5x2andnx=2x+5x Write an equation for the inverse function for fx=4x+3.Write an equation for the inverse function for the one-to-one function defined by fx=x2x+2.Given nx=x2+1forx0, write an equation of the inverse.Given gx=x+2, find an equation of the inverse.Given the function f=1,2,2,3,3,4 write the set of ordered pairs representing f1 .The graphs of a function and its inverse are symmetric with respect to the line .If no horizontal line intersects the graph of a function f in more than one point, then f is a -- function.Given a one-to one function f, if fa=fb,thenab.Let f be a one-to-one function and let g be the inverse of f . Then fgx=andgfx=.If a,b is a point on the graph of a one-to-one function f, then the corresponding ordered pair is a point on the graph of f1 .For Exercises 7-12, a relation in x and y is given. Determine if the relation defined y as a one-to-one function of x. (See Example 1) 6,5,4,2,3,1,8,4 For Exercises 7-12, a relation in x and y is given. Determine if the relation defined y as a one-to-one function of x. (See Example 1) 14,1,2,3,7,4,9,2 For Exercises 7-12, a relation in x and y is given. Determine if the relation defined y as a one-to-one function of x. (See Example 1)For Exercises 7-12, a relation in x and y is given. Determine if the relation defined y as a one-to-one function of x. (See Example 1)For Exercises 7-12, a relation in x and y is given. Determine if the relation defined y as a one-to-one function of x. (See Example 1)For Exercises 7-12, a relation in x and y is given. Determine if the relation defined y as a one-to-one function of x. (See Example 1)For Exercises 13-22, determine if the relation defined y as a one-to-one function of x. (See Example 2)For Exercises 13-22, determine if the relation defined y as a one-to-one function of x. (See Example 2)For Exercises 13-22, determine if the relation defined y as a one-to-one function of x. (See Example 2)For Exercises 13-22, determine if the relation defined y as a one-to-one function of x. (See Example 2)For Exercises 13-22, determine if the relation defined y as a one-to-one function of x. (See Example 2)For Exercises 13-22, determine if the relation defined y as a one-to-one function of x. (See Example 2)For Exercises 13-22, determine if the relation defined y as a one-to-one function of x. (See Example 2)For Exercises 13-22, determine if the relation defined y as a one-to-one function of x. (See Example 2)For Exercises 13-22, determine if the relation defined y as a one-to-one function of x. (See Example 2)For Exercises 13-22, determine if the relation defined y as a one-to-one function of x. (See Example 2)For Exercises 23-30, use the definition of a one-to-one function to determine if the function is one-to-one. (See Example 3) fx=4x7 For Exercises 23-30, use the definition of a one-to-one function to determine if the function is one-to-one. (See Example 3) hx=3x+2 For Exercises 23-30, use the definition of a one-to-one function to determine if the function is one-to-one. (See Example 3) gx=x3+8 For Exercises 23-30, use the definition of a one-to-one function to determine if the function is one-to-one. (See Example 3) kx=x327 For Exercises 23-30, use the definition of a one-to-one function to determine if the function is one-to-one. (See Example 3) mx=x24 For Exercises 23-30, use the definition of a one-to-one function to determine if the function is one-to-one. (See Example 3) nx=x2+1 For Exercises 23-30, use the definition of a one-to-one function to determine if the function is one-to-one. (See Example 3) px=x+1 For Exercises 23-30, use the definition of a one-to-one function to determine if the function is one-to-one. (See Example 3) qx=x3 For Exercises 31-36, determine whether the two functions are inverses. (See Example 4) fx=5x+4andgx=x45 For Exercises 31-36, determine whether the two functions are inverses. (See Example 4) hx=7x3andkx=x+37 For Exercises 31-36, determine whether the two functions are inverses. (See Example 4) mx=2+x6andnx=6x2 For Exercises 31-36, determine whether the two functions are inverses. (See Example 4) px=3+x4andqx=4x3 For Exercises 31-36, determine whether the two functions are inverses. (See Example 4) tx=4x1andvx=x+4x For Exercises 31-36, determine whether the two functions are inverses. (See Example 4) wx=6x+2andzx=62xx There were 2000 applicants for enrollment to the freshman class at a small college in the year 2010 The number of applications has risen linearly by roughly 150 per year. The number of applications fx is given by fx=2000+150x, where x is the number of years since 2010. a. Determine if the function gx=x2000150 is the inverse of f . b. Interpret the meaning of function g in the context of this problem.The monthly sales for January for a whole foods market was $60,000 and has increased linearly by $2500 per month. The amount in sales fxin$ is given by fx=60,000+2500x, where x is the number of months since January. a. Determine if the function gx=x60,0002500 is the inverse of f . b. Interpret the meaning of function g in the context of this problem.a. Show that fx=2x3 defines a one-to-one function. b. Write an equation for f1x. c. Graph y=fxandy=f1x on the same coordinate system.a. Show that fx=4x+4 defines a one-to-one function. b. Write an equation for f1x. c. Graph y=fxandy=f1x on the same coordinate system.For Exercises 41-52, a one-to-one function is given. Write an equation for the inverse function. (See Example 5-6) fx=4x9 For Exercises 41-52, a one-to-one function is given. Write an equation for the inverse function. (See Example 5-6) gx=8x3 For Exercises 41-52, a one-to-one function is given. Write an equation for the inverse function. (See Example 5-6) hx=x53 For Exercises 41-52, a one-to-one function is given. Write an equation for the inverse function. (See Example 5-6) kx=x+83 For Exercises 41-52, a one-to-one function is given. Write an equation for the inverse function. (See Example 5-6) mx=4x3+2 For Exercises 41-52, a one-to-one function is given. Write an equation for the inverse function. (See Example 5-6) nx=2x35 For Exercises 41-52, a one-to-one function is given. Write an equation for the inverse function. (See Example 5-6) cx=5x+2 For Exercises 41-52, a one-to-one function is given. Write an equation for the inverse function. (See Example 5-6) sx=2x3 For Exercises 41-52, a one-to-one function is given. Write an equation for the inverse function. (See Example 5-6) tx=x4x+2 For Exercises 41-52, a one-to-one function is given. Write an equation for the inverse function. (See Example 5-6) vx=x5x+1 For Exercises 41-52, a one-to-one function is given. Write an equation for the inverse function. (See Example 5-6) fx=xa3bc For Exercises 41-52, a one-to-one function is given. Write an equation for the inverse function. (See Example 5-6) gx=bx+a3+c a. Graph fx=x23;x0. (See Example 7) b. From the graph of f , is f a one-to-one function? c. Write the domain of f in interval notation. d. Write the range of f in interval notation. e. Write an equation for f1x. f. Graph y=fx andy=f1x on the same coordinate system. g. Write the domain of f1 in interval notation. h. Write the range of f1 in interval notation.a. Graph fx=x2+1;x0. b. From the graph of f , is f a one-to-one function? c. Write the domain of f in interval notation. d. Write the range of f in interval notation. e. Write an equation for f1x. f. Graph y=fx andy=f1x on the same coordinate system. g. Write the domain of f1 in interval notation. h. Write the range of f1 in interval notation a. Graph fx=x+1. (See Example 8) b. From the graph of f , is f a one-to-one function? c. Write the domain of f in interval notation. d. Write the range of f in interval notation. e. Write an equation for f1x. f. Explain why the restriction x0 is placed on f1 . g. Graph y=fx andy=f1x on the same coordinate system. h. Write the domain of f1 in interval notation. i. Write the range of f1 in interval notation.a. Graph fx=x2. b. From the graph of f , is f a one-to-one function? c. Write the domain of f in interval notation. d. Write the range of f in interval notation. e. Write an equation for f1x. f. Explain why the restriction x0 is placed on f1 . g. Graph y=fx andy=f1x on the same coordinate system. h. Write the domain of f1 in interval notation. i. Write the range of f1 in interval notation.Given that the domain of a one-to-one function f is 0, and the range of f is 0,4 , state the domain and range of f1 .Given that the domain of a one-to-one function f is 3,5 and the range of f is 2, , state the domain and range of f1 .Given fx=x+3;x0, write an equation for f1 .Given fx=x3;x0, write an equation for f1 .For Exercises 61-66, fill in the blanks and determine an equation for f1x mentally. If function f adds 6 to x, then f16 from x. Function f is defined by fx=x+6, and function f1 is defined by f1x=.For Exercises 61-66, fill in the blanks and determine an equation for f1x mentally. If function f multiplies x by 2, then f1x by 2. Function f is defined by fx=2x, and function f1 is defined by f1x=.For Exercises 61-66, fill in the blanks and determine an equation for f1x mentally. Suppose that function f multiplies x by 7 and subtracts 4. Write an equation for f1x .For Exercises 61-66, fill in the blanks and determine an equation for f1x mentally. Suppose that function f divides x by 3 and adds 11. Write an equation for f1x .For Exercises 61-66, fill in the blanks and determine an equation for f1x mentally. Suppose that function f cubes x and adds 20. Write an equation for f1x .For Exercises 61-66, fill in the blanks and determine an equation for f1x mentally. Suppose that function f takes the cube root of x and subtracts 10. Write an equation for f1x .For Exercises 67-70, find the inverse mentally. fx=8x+1 For Exercises 67-70, find the inverse mentally. px=2x10 For Exercises 67-70, find the inverse mentally. qx=x45+1 For Exercises 67-70, find the inverse mentally. mx=4x3+3 For Exercises 71-74, the graph of a function is given. Graph the inverse function.For Exercises 71-74, the graph of a function is given. Graph the inverse function.For Exercises 71-74, the graph of a function is given. Graph the inverse function.74PE For Exercises 75-76, the table defines Y1=fx as a one-to-one function of x. Find the values of f1 for the selected values of x. a.f132b.f12.5c.f126 For Exercises 75-76, the table defines Y1=fx as a one-to-one function of x. Find the values of f1 for the selected values of x. a.f15b.f19.45c.f18 For Exercises 77-80, determine if the statement is true or false. If a statement is false, explain why. All linear functions with a nonzero slope have an inverse function.For Exercises 77-80, determine if the statement is true or false. If a statement is false, explain why. The domain of any one-to-one function is the same as the domain of its inverse function.For Exercises 77-80, determine if the statement is true or false. If a statement is false, explain why. The range of a one-to-one function is the same as the range of its inverse function.For Exercises 77-80, determine if the statement is true or false. If a statement is false, explain why. No quadratic function defined by fx=ax2+bx+ca0 is one-to-one.Based on data from Hurricane Katrina, the function defined by wx=1.17x+1220 gives the wind speed wx (in mph) based on the barometric pressure x (in millibars, mb). a. Approximate the wind speed for a hurricane with a barometric pressure of 1000 mb. b. Write a function representing the inverse of w and interpret its meaning in context. c. Approximate the barometric pressure for a hurricane with wind speed 100 mph. Round to the nearest mb.The function defined by Fx=95x+32 gives the temperature Fx (in degrees Fahrenheit) based on the temperature x (in Celsius). a. Determine the temperature in Fahrenheit if the temperature in Celsius is 25oC. b. Write a function representing the inverse of F and interpret its meaning in context. c. Determine the temperature in Celsius if the temperature in Fahrenheit is 5F.Suppose that during normal respiration, the volume of air inhaled per breath (called â€œtidal volumeâ€�) by a mammal of any size is 6.33 mL per kilogram of body mass. a. Write a function representing the tidal volume Tx (in mL) of a mammal of mass x (in kg). b. Write an equation for T1x. c. What does the inverse function represent in the context of this problem? d. Find T1170 and interpret its meaning in context. Round to the nearest whole unit.At a cruising altitude of 35,000 ft, a certain airplane travels 555 mph. a. Write a function representing the distance dx (in mi) for x hours at cruising altitude. b. Write an equation for d1x . c. What does the inverse function represent in the Context of lit problem? d. Evaluate d12553 and interpret its meaning in context.The millage rate is the amount of property tax per $1000 of the taxable value of a home. For a certain county the millage rate is 24 mil ($24 in tax per $1000 of taxable value of the home). A city within the county also imposes a flat fee of $108 per home. a. Write a function representing the total amount of property tax Txin$ for a home with a taxable value of x thousand dollars. b. Write an equation for T1x . c. What does the inverse function represent in the context of this problem? d. Evaluate T12988 and interpret its meaning in context Beginning on January 1, park rangers in Everglades National Park began recording the water level for one particularly dry area of the park. The water level was initially 2.5 ft and decreased by approximately 0.015 ft/day. a. Write a function representing the water level Lx (in ft), x days after January 1. b. Write an equation for L1x . c. What does the inverse handier, represent in the context of this problem? d. Evaluate L11.9 and interpret its meaning in context.87PE 88PE Explain why if a horizontal line intersects the graph of a function in more than one point, then the function is not one-to-one.90PE 91PE Consider a function defined as follows: Given x, the value fx is the exponent above the base of 3 that produces x. For example, f9=2 because 32=9. Evaluate a.f27b.f81c.f3d.f19 93PE A function is said to be periodic if there exists some nonzero real number p, called the period, such that fx+p=fx for all real number x in the domain of f. Explain why no periodic function is one-to-one.Graph the functions. a.fx=5xb.gx=15x Graph. gx=2x+21 Explain how the graph of fx=ex1 is related to the graph of y=ex.Suppose that $8000 is invested and pays 4.5% per year under the following compounding options. a. Compounded annually b. Compounded quarterly c. Compounded monthly d. Compounded daily e. Compounded continuously Determine the total amount in the account after 5 yr with each option.Cesium-137 is a radioactive metal with a short half-life of 30 yr. In a sample originally having 2 g of cesium-137, the amount At (in grams) of cesium-137 present after t years is given by At=212t/30. How much cesium-137 will be present after a. 30 yr? b. 60 yr? c. 90 yr?The function defined by y=x3 (is/is not) an exponential function, whereas the function defined by y=3x (is/is not) an exponential function.The graph of fx=53x is (increasing/decreasing) over its domain.The graph of fx=35x is (increasing/decreasing) over its domain.The domain of an exponential function fx=bxis.The range of an exponential function fx=bxis.All exponential functions fx=bx pass through the point .The horizontal asymptote of an exponential function fx=bx is the line .As x, the value of 1+1xx approaches .For Exercises 9-12, evaluate the functions the given values of x. Round to 4 decimal places if necessary. fx=5xa.f1b.f4.8c.f2d.f For Exercises 9-12, evaluate the functions the given values of x. Round to 4 decimal places if necessary. gx=7xa.g2b.g5.9c.g11d.ge For Exercises 9-12, evaluate the functions the given values of x. Round to 4 decimal places if necessary. hx=14xa.h3b.h1.4c.h3d.h0.5e For Exercises 9-12, evaluate the functions the given values of x. Round to 4 decimal places if necessary. kx=16xa.k3b.k1.4c.k0.5d.k0.5 Which function are exponential functions? a.fx=4.2xb.gx=x4.2c.hx=4.2xd.kx=4.2xe.mx=4.2x Which function are exponential functions? a.vx=xb.tx=xc.wx=xd.nx=xe.px=x For Exercises 15-22, graph the functions and write the domain and range in interval notation. (See Example 1) fx=3x For Exercises 15-22, graph the functions and write the domain and range in interval notation. (See Example 1) gx=4x For Exercises 15-22, graph the functions and write the domain and range in interval notation. (See Example 1) hx=13x For Exercises 15-22, graph the functions and write the domain and range in interval notation. (See Example 1) kx=14x For Exercises 15-22, graph the functions and write the domain and range in interval notation. (See Example 1) mx=32x For Exercises 15-22, graph the functions and write the domain and range in interval notation. (See Example 1) nx=54x For Exercises 15-22, graph the functions and write the domain and range in interval notation. (See Example 1) bx=23x For Exercises 15-22, graph the functions and write the domain and range in interval notation. (See Example 1) cx=45x For Exercises 23-32, a. Use transformations of the graphs of y=3x (See Exercise 15) and y=4x (See Exercise 16) to graph the given function. (See Example 2) b. Write the domain and range in interval notation. c. Write an equation of the asymptote. fx=3x+2 For Exercises 23-32, a. Use transformations of the graphs of y=3x (See Exercise 15) and y=4x (See Exercise 16) to graph the given function. (See Example 2) b. Write the domain and range in interval notation. c. Write an equation of the asymptote. gx=4x3 For Exercises 23-32, a. Use transformations of the graphs of y=3x (See Exercise 15) and y=4x (See Exercise 16) to graph the given function. (See Example 2) b. Write the domain and range in interval notation. c. Write an equation of the asymptote. mx=3x+2 26PE For Exercises 23-32, a. Use transformations of the graphs of y=3x (See Exercise 15) and y=4x (See Exercise 16) to graph the given function. (See Example 2) b. Write the domain and range in interval notation. c. Write an equation of the asymptote. px=3x41 For Exercises 23-32, a. Use transformations of the graphs of y=3x (See Exercise 15) and y=4x (See Exercise 16) to graph the given function. (See Example 2) b. Write the domain and range in interval notation. c. Write an equation of the asymptote. qx=4x+1+2 For Exercises 23-32, a. Use transformations of the graphs of y=3x (See Exercise 15) and y=4x (See Exercise 16) to graph the given function. (See Example 2) b. Write the domain and range in interval notation. c. Write an equation of the asymptote. kx=3x For Exercises 23-32, a. Use transformations of the graphs of y=3x (See Exercise 15) and y=4x (See Exercise 16) to graph the given function. (See Example 2) b. Write the domain and range in interval notation. c. Write an equation of the asymptote. hx=4x For Exercises 23-32, a. Use transformations of the graphs of y=3x (See Exercise 15) and y=4x (See Exercise 16) to graph the given function. (See Example 2) b. Write the domain and range in interval notation. c. Write an equation of the asymptote. tx=3x For Exercises 23-32, a. Use transformations of the graphs of y=3x (See Exercise 15) and y=4x (See Exercise 16) to graph the given function. (See Example 2) b. Write the domain and range in interval notation. c. Write an equation of the asymptote. vx=4x For Exercises 33-36, a. Use transformations of the graphs of y=13x (See Exercise 17) and y=14x (See Exercise 18) to graph the given function. (See Example 2) b. Write the domain and range in interval notation. c. Write an equation of the asymptote. fx=13x+13 For Exercises 33-36, a. Use transformations of the graphs of y=13x (See Exercise 17) and y=14x (See Exercise 18) to graph the given function. (See Example 2) b. Write the domain and range in interval notation. c. Write an equation of the asymptote. gx=14x2+1 35PE 36PE For Exercises 37-38, evaluate the functions for the given values of x. Round to 4 decimal places. fx=exa.f4b.f3.2c.f13d.f 38PE 39PE For Exercises 39-44, a. Use transformations of the graph of y=ex to graph the given function. (See Example 3) b. Write the domain and range in interval notation. c. Write an equation of the asymptote. gx=ex2 For Exercises 39-44, a. Use transformations of the graph of y=ex to graph the given function. (See Example 3) b. Write the domain and range in interval notation. c. Write an equation of the asymptote. hx=ex+2 42PE For Exercises 39-44, a. Use transformations of the graph of y=ex to graph the given function. (See Example 3) b. Write the domain and range in interval notation. c. Write an equation of the asymptote. mx=ex3 For Exercises 39-44, a. Use transformations of the graph of y=ex to graph the given function. (See Example 3) b. Write the domain and range in interval notation. c. Write an equation of the asymptote. nx=ex+4 For Exercises 45-46, complete the table to determine the effect of the number of compounding periods when computing interest. (See Example 4) Suppose that $10,000 is invested at 4 interest for 5 yr under the following compounding options. Complete the table.For Exercises 45-46, complete the table to determine the effect of the number of compounding periods when computing interest. (See Example 4) Suppose that $8000 is invested at 3.5 interest for 20 yr under the following compounding options. Complete the table.47PE For Exercises 47-48, suppose that P dollars in principal is invested for t years at the given interest rates with continuous compounding. Determine the amount that the investment is worth at the end of the end of the given time period. P=$6000,t=12yra.1interestb.2interestc.4.5interest 49PE Al needs to borrow $15,000 to buy a car. He can borrow the money at 6.7 simple interest for 5 yr or he can borrow at 6.4 interest compounded continuously for 5 yr. a. How much total interest would A1 pay at 6.7 simple interest? b. How much total interest would A1 pay at 6.4 interest compounded continuously? c. Which option results is less total interest?Jerome wants to invest $25,000 as part of his retirement plan. He can invest the money at 5.2 simple interest for 30 yr, or he can invest at 3.8 interest compounded continuously for 30 yr. Which option results in more total interest?Heather wants to invest $35,000 of her retirement. She can invest at 4.8 simple interest for 20 yr, or she can choose an option with 3.6 interest compounded continuously for 20 yr. Which option results in more total interest?Strontium-90 90Sr is a by-product of nuclear fission with a half-life of approximately 28.9 yr. After the Chernobyl nuclear reactor accident in 1986, large areas surrounding the site were contaminated with 90Sr . if 10g (micrograms) of 90Sr is present in a sample, the function At=1012t/28.9 gives the amount Ating present after t years. Evaluate the function for the given values of t and interpret the meaning in context. Round to 3 decimal places if necessary. (See Example 5) a.A28.9b.A57.8c.A100 In 2006, the murder of Alexander Litvinenko a Russian dissident, was through to be by poisoning from the rare and highly radioactive element poloninum-210 210Po. The half-life of 210Po is 138.4 yr. If 0.1 mg of 210Po is present in a sample then At=0.112t/138.4 gives the amount Atinmg present after t years. Evaluate the function for the given values of t and interpret the meaning in context. Round to 3 decimal places if necessary. a.A138.4b.A276.8c.A500 According to the CIAâ€™s World Fact Book in 2010, the population of the United Stales was approximately 310 million with a 0.97 annual growth rate. At this rate, the population Pt (in millions) can be approximated by Pt=3101.0097t, where t is the time in years since 2010. a. Is the graph of P an increasing or decreasing exponential function? b. Evaluate P0 and interpret its meaning in the context of this problem. c. Evaluate P10 and interpret its meaning in the context of this problem Round the population value to the nearest million. d. Evaluate P20 and P30 . e. Evaluate P200 and use this result to determine if it is reasonable to expect this model to continue indeiritely.The population of Canada in 2010 was approximately 34 million with an annual growth rate of 0.804 . At this rate, the population Pt (in millions) can be approximated by Pt=341.00804t, where t is the time in years since 2010. a. Is the graph of P an increasing or decreasing exponential function? b. Evaluate P0 and interpret its meaning in the context of this problem. c. Evaluate P5 and interpret its meaning in the context of this problem Round the population value to the nearest million. d. Evaluate P15 and P25 .The atmospheric pressure on an object decreases as altitude increases. If a is the height (in km) above sea heel, then the pressure Pa (in mmHg) is approximated by Pa=760e0.13a. a. Find the atmospheric pressure at sea level. b. Determine the atmospheric pressure at 8.848 km (the altitude of Mt. Everest). Round to the nearest whole unit.The function defined by At=100e0.0318t approximates the equivalent amount of money needed t years after the year 2010 to equal $100 of buying power in the year 2010. The value 0.0318 is related to the average rate of inflation. a. Evaluate A15 and interpret its meaning in the context of this problem b. Verify that by the year 2032, more than $200 will be needed to have the same buying power as $100 in 2010.Newton's law of cooling Indicates that the temperature of a warm object, such as a cake coming out of the oven, will decrease exponentially with time and will approach the temperature of the surrounding air. The temperature Tt is modeled by Tt=Ta+T0Taekt. In this model. Ta represents the temperature of the surrounding air, T0 represents the initial temperature of the object, and t is the dine after the object rearm cooling. The value of k is a constant of proportion relating the temperature of the object to its rate of temperature change. Use this model for Exercises 59-60. A cake comes out of the oven at 350oF and is placed on a cooling rack in a 78oF kitchen. After checking the temperature several minutes later, the value of k is measured as 0.046. a. Write a function that models the temperature TtinoF of the cake t minutes after being removed from the oven. b. What is the temperature of the cake 10 min after coming out of the oven? Round to the nearest degree. c. It is recommended that the cake should not be frosted until it has cooled to under 100F . If Jessica waits 1 hr to frost the cake, will the cake be cool enough to frost?Newton's law of cooling Indicates that the temperature of a warm object, such as a cake coming out of the oven, will decrease exponentially with time and will approach the temperature of the surrounding air. The temperature Tt is modeled by Tt=Ta+T0Taekt. In this model. Ta represents the temperature of the surrounding air, T0 represents the initial temperature of the object, and t is the dine after the object rearm cooling. The value of k is a constant of proportion relating the temperature of the object to its rate of temperature change. Use this model for Exercises 59-60. Water in a water heater is originally 122F . The water heater is shut off and the water cools to the temperature of the surrounding air, which is 60F . The water cools slowly because of the insulation inside the heater, and the value of k is measured as 0.00351. a. Write a function that models the temperature TtinoF of the water t hours after the water heater is Shut off. b. what is the temperature of the water 12 hr after the heater is shut off? Round to the nearest degree. c. Dominic does not like to shower with water less than 115F . If Dominic mho 24 hr, will the water still be warm enough for a shower?A farmer depreciates a $120,000 tractor. He estimates that the resale value Vtin$1000 of the tractor t years after purchase is 80 of its value from the previous year. Therefore, the resale value can be approximated by Vt=1200.8t. a. Find the resale value 5 yr after purchase. Round to the nearest $1000 . b. The farmer estimates that the cost to run the tractor is $18/hr in labor, $36/hr in fuel, and $22/hr in overhead colts (for maintenance and repair). Estimate the farmer's cost to run the tractor for the first year if he runs the tractor for a total of 800 hr. Include hourly costs and depreciation.A veterinarian depreciates a $10,000 X-ray machine. He estimates that the resale value Vtin$ after t years is 90 of its value from the previous year. Therefore, the resale value can be approximated by Vt=10,0000.9t. a. Find the resale value after 4 yr. b. If the veterinarian wants to sell his practice 8 yr after the X-ray machine was purchased. how much is the machine worth? Round to the nearest $100 .For Exercises 63-64, solve the equation in parts ac by inspection. Then estimate the solutions to parts (d) and â‚¬ between two consecutive integers. a.2x=4b.2x=8c.2x=16d.2x=7e.2x=10 For Exercises 63-64, solve the equation in parts ac by inspection. Then estimate the solutions to parts (d) and â‚¬ between two consecutive integers. a.3x=3b.3x=9c.3x=27d.3x=7e.3x=10 a. Graph fx=2x. (See Example 1) b. Is f a one-to-one function? c. Write the domain and range of f in interval notation. d. Graph f1 on the same coordinate system as f. e. Write the domain and range of f1 in interval notation. f. From the graph evaluate f1 (1), f1 (2), and f1 (4).66PE Refer to the graphs of fx=2x and the inverse function, y=f1x from Exercise 65. Fill in the blanks. a.Asx,fx.b.Asx,fx.c.Asx,f1x.d.Asx0+,f1x.Refer to the graphs of gx=3x and the inverse function, y=g1x from Exercise 65. Fill in the blanks. a.Asx,gx.b.Asx,gx.c.Asx,g1x.d.Asx0+,g1x.Explain why the equation 2x=2 has no solution.Explain why the fx=x2 is not an exponential function.For Exercises 71-72, find the real solutions to the equation. 3x2ex6xex=0 For Exercises 71-72, find the real solutions to the equation. x2exex=0 Use the properties of exponents to simplify. a.exehb.ex2c.exehd.exexe.e2x Factor. a.ex+hexb.e4xe2x Multiply. ex+ex2 Multiply. exex2 Show that ex+ex22exex22=1.78PE 79PE For Exercises 79-80, find the difference quotient fx+hfxh. write the answers in factored form. fx=2x Graph the following functions on the window 3,3,1by1,8,1 and comment on the behaviour of the graphs near x=0. Y1=exY2=1+x+x22Y3=1+x+x22+x36 Write each equation in exponential form. a.log39=2b.log1011000=3c.log61=0 Write each equation in logarithmic form. a.25=32b.104=10,000c.182=64 Evaluate each expression. a.log5125b.log381c.log4164 Evaluate. a.log10,000,000b.log0.1c.lne5d.lne Approximate the logarithms. Round to 4 decimal places. a.log229b.log3.761012c.log0.0216d.ln87e.ln0.0032f.ln Simplify. a.log1313b.lnec.aloga3d.eln6e.log1f.log1g.log992h.log10e Graph the functions. a.y=log4xb.y=log1/2x Graph the function. Identify the vertical asymptote and write the domain in interval notation. gx=log3x4+1 Write the domain in interval notation. a.log413xb.log2+xc.mx=ln64x2 a. Determine the magnitude of an earthquake that is 105.2 times I0. b. Determine the magnitude of an earthquake that is 104.2 times I0. c. How many times more intense is a 5.2-magnitude earthquake than a 4.2-magnitude earthquake?Given positive real numbers x and b such that b1,y=logbx is the function base b and is equivalent to by=x.Given y=logbx, the value y is called the , b is called the , and x is called the .The logarithmic function base 10 is called the logarithmic function, and the logarithmic function base e is called the logarithmic function.Given y=logx, the base is understood to be . Given y=lnx, the base is understood to be .logb1= because b=1.logbb= because b=b.f(x)=logbxandg(x)=bx are inverse functions Therefore, logbbx=andblogbx=.The graph of y=logbx passes through the point (1, 0) and the line is a (horizontal/vertical) asymptote.For Exercises 9-16, write the equation in exponential form. (See Example 1) log864=2 For Exercises 9-16, write the equation in exponential form. (See Example 1) log981=2 For Exercises 9-16, write the equation in exponential form. (See Example 1) log110,000=4 For Exercises 9-16, write the equation in exponential form. (See Example 1) log11,000,000=6 For Exercises 9-16, write the equation in exponential form. (See Example 1) In1=0 For Exercises 9-16, write the equation in exponential form. (See Example 1) log81=0 15PE 16PE 17PE For Exercises 17-24, write the equation in logarithmic form. (See Example 2) 25=32 19PE For Exercises 17-24, write the equation in logarithmic form. (See Example 2) 125=32 21PE For Exercises 17-24, write the equation in logarithmic form. (See Example 2) e1=e 23PE For Exercises 17-24, write the equation in logarithmic form. (See Example 2) M3=N 25PE For Exercises 25-50, simplify the expression without using a calculator. (See Examples 3-4) log216 27PE 28PE 29PE For Exercises 25-50, simplify the expression without using a calculator. (See Examples 3-4) log10,000,000 31PE For Exercises 25-50, simplify the expression without using a calculator. (See Examples 3-4) log319 33PE For Exercises 25-50, simplify the expression without using a calculator. (See Examples 3-4) log110,000 35PE For Exercises 25-50, simplify the expression without using a calculator. (See Examples 3-4) lne10 37PE For Exercises 25-50, simplify the expression without using a calculator. (See Examples 3-4) ln1e8 39PE For Exercises 25-50, simplify the expression without using a calculator. (See Examples 3-4) log1/416 41PE For Exercises 25-50, simplify the expression without using a calculator. (See Examples 3-4) log1/6136 For Exercises 25-50, simplify the expression without using a calculator. (See Examples 3-4) log0.00001 For Exercises 25-50, simplify the expression without using a calculator. (See Examples 3-4) log0.0001 45PE For Exercises 25-50, simplify the expression without using a calculator. (See Examples 3-4) log3/294 For Exercises 25-50, simplify the expression without using a calculator. (See Examples 3-4) log335 48PE 49PE 50PE For Exercises 51-52, estimate the value of each logarithm between two consecutive integers. Then use a calculator to approximate the value to 4 decimal places. For example, log8970 is between 3 and 4 because 1038970104. (See Example 5) a.log46,832b.log1,247,310c.log0.24d.log0.0000032e.log5.6105f.log5.1103 For Exercises 51-52, estimate the value of each logarithm between two consecutive integers. Then use a calculator to approximate the value to 4 decimal places. For example, log8970 is between 3 and 4 because 1038970104. (See Example 5) a.log293,416b.log897c.log0.038d.log0.00061e.log9.1108f.log8.2102 53PE For Exercises 53-54, approximate fx=lnx for the given values of x. Round to 4 decimal places. (See Example 5) a.f1860b.f0.0694c.f87d.f2e.f1.31012f.f8.51017 For Exercises 55-64, simplify the expression without using a calculator. (See Example 6) log4411 For Exercises 55-64, simplify the expression without using a calculator. (See Example 6) log667 For Exercises 55-64, simplify the expression without using a calculator. (See Example 6) logcc For Exercises 55-64, simplify the expression without using a calculator. (See Example 6) logdd 59PE For Exercises 55-64, simplify the expression without using a calculator. (See Example 6) 4log4ac For Exercises 55-64, simplify the expression without using a calculator. (See Example 6) lnea+b For Exercises 55-64, simplify the expression without using a calculator. (See Example 6) lnex2+1 For Exercises 55-64, simplify the expression without using a calculator. (See Example 6) log51 For Exercises 55-64, simplify the expression without using a calculator. (See Example 6) log1 For Exercises 65-70, graph the function. (See Example 7) y=log3x For Exercises 65-70, graph the function. (See Example 7) y=log5x 67PE 68PE 69PE 70PE 71PE 72PE 73PE 74PE 75PE For Exercises 71-78, (See Example 8) a. Use transformations of the graphs of y=log2x(SeeExample7)andy=log3x (see Exercise 65) to graph the given functions. b. Write the domain and range in interval notation. c. Write an equation of the asymptote. y=log2x21 77PE 78PE 79PE For Exercises 79-92, write the domain in interval notation. (See Example 9) gx=log3x For Exercises 79-92, write the domain in interval notation. (See Example 9) hx=log26x+7 For Exercises 79-92, write the domain in interval notation. (See Example 9) kx=log35x+6 83PE 84PE 85PE 86PE For Exercises 79-92, write the domain in interval notation. (See Example 9) mx=3+ln111x For Exercises 79-92, write the domain in interval notation. (See Example 9) nx=4log1x+5 89PE For Exercises 79-92, write the domain in interval notation. (See Example 9) qx=logx2+10x+9 91PE 92PE 93PE The intensities of earthquakes are measured with seismographs all over the world at different distances from the epicenter. Suppose that the intensity of a medium earthquake is originally reported as 105.4timesI0. Later this value is revised as 105.8timesI0. a. Determine the magnitude of the earthquake using the original estimate for intensity. b. Determine the magnitude using the revised estimate for intensity. c. How many times more intense was the earthquake than originally thought? Round to 1 decimal place.Sounds are produced when vibrating objects create pressure waves in some medium such as air. When these variations in pressure reach the human eardrum, it causes the eardrum to vibrate in a similar manner and the ear detects sound. The intensity of sound is measured as power per unit area. The threshold for hearing (minimum sound detectable by a young, healthy ear) is defined to be I0=1012W/m2 (watts per square meter). The sound level L, or â€œloudnessâ€� of sound, is measured in decibels (dB) as L=10logII0, where I is the intensity of the given sound. Use this formula for Exercises 95-96. a. Find the sound level of a jet plane taking off if its intensity is 1015 times the intensity of I0. b. Find the sound level of the noise from city traffic if its intensity is 109 times I0. c. How many times more intense is the sound of a jet plane taking off than noise from city traffic?Sounds are produced when vibrating objects create pressure waves in some medium such as air. When these variations in pressure reach the human eardrum, it causes the eardrum to vibrate in a similar manner and the ear detects sound. The intensity of sound is measured as power per unit area. The threshold for hearing (minimum sound detectable by a young, healthy ear) is defined to be I0=1012W/m2 (watts per square meter). The sound level L, or â€œloudnessâ€� of sound, is measured in decibels (dB) as L=10logII0, where I is the intensity of the given sound. Use this formula for Exercises 95-96. a. Find the sound level of a motorcycle if its intensity is 1010timesI0. b. Find the sound level of a vacuum cleaner if its intensity is 107timesI0. c. How many times more intense is the sound of a motorcycle than a vacuum cleaner?Scientists use the pH scale to represent the level of acidity or alkalinity of a liquid. This is based on the molar concentration of hydrogen ions, [H+]. Since the values of [H+] vary over a large range, 1100 mole per liter to 11014 mole per liter mol/L, a logarithmic scale is used to compute pH. The formula pH=logH+ represents the pH of a liquid as a function of its concentration of hydrogen ions, [H+]. The pH scale ranges from 0 to 14. Pure water is taken as neutral having a pH of 7. A pH less than 7 is acidic. A pH greater than 7 is alkaline (or basic). For Exercises 97-98, use the formula for pH. Round pH values to 1 decimal place. Vinegar and lemon juice are both acids. Their [H+] values are 5.0103mol/Land1102mol/L, respectively. a. Find the pH for vinegar. b. Find the pH for lemon juice. c. Which substance is more acidic?Scientists use the pH scale to represent the level of acidity or alkalinity of a liquid. This is based on the molar concentration of hydrogen ions, [H+]. Since the values of [H+] vary over a large range, 1100 mole per liter to 11014 mole per liter mol/L, a logarithmic scale is used to compute pH. The formula pH=logH+ represents the pH of a liquid as a function of its concentration of hydrogen ions, [H+]. The pH scale ranges from 0 to 14. Pure water is taken as neutral having a pH of 7. A pH less than 7 is acidic. A pH greater than 7 is alkaline (or basic). For Exercises 97-98, use the formula for pH. Round pH values to 1 decimal place. Bleach and milk of magnesia are both bases. Their H+ values are 2.01013mol/Land4.11010mol/L, respectively. a. Find the pH for bleach. b. Find the pH for milk of magnesia. c. Which substance is more basic?99PE For Exercises 99-102, a. Write the equation in exponential form. b. Solve the equation from part (a). c. Verify that the solution checks in the original equation. log2x5=4 For Exercises 99-102, a. Write the equation in exponential form. b. Solve the equation from part (a). c. Verify that the solution checks in the original equation. log47x6=3 For Exercises 99-102, a. Write the equation in exponential form. b. Solve the equation from part (a). c. Verify that the solution checks in the original equation. log59x11=2 For Exercises 103-106, evaluate the expressions. log3log464 For Exercises 103-106, evaluate the expressions. log2log1/214 For Exercises 103-106, evaluate the expressions. log16log813

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