Precalculus Graphs And Models A Right Triangle Approach 6th Edition by Marvin L. Bittinger – Test Bank

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ExamName___________________________________MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.Match the function with one of the graphs.1)f(x)=1+ex1)A)B)C)D)Answer:BExplanation:A)B)C)D)1
2)f(x)=-14×2)A)B)C)D)Answer:AExplanation:A)B)C)D)2
Choose the function that might be used as a model for the data in the scatter plot.3)3)A)Logarithmic, f(x)=a+b ln xB)Exponential, f(x)=ab-xor f(x)=P0e-kx, k>0C)Quadratic, f(x)=ax2+bx+cD)Exponential, f(x)=abxor f(x)=P0ekx, k>0Answer:DExplanation:A)B)C)D)3
Match the function with one of the graphs.4)f(x)=ex-54)A)B)C)D)Answer:CExplanation:A)B)C)D)4
5)f(x)=2e-x5)A)B)C)D)Answer:CExplanation:A)B)C)D)5
Choose the function that might be used as a model for the data in the scatter plot.6)6)A)Polynomial, not quadraticB)Exponential, f(x)=ab-xor f(x)=P0e-kx, k>0C)Logistic, f(x)=a1+be-kxD)Quadratic, f(x)=ax2+bx+cAnswer:AExplanation:A)B)C)D)7)7)A)Logarithmic, f(x)=a+b ln xB)Polynomial, not quadraticC)Exponential, f(x)=ab-xor f(x)=P0e-kx, k>0D)Exponential, f(x)=abxor f(x)=P0ekx, k>0Answer:CExplanation:A)B)C)D)6
Match the function with one of the graphs.8)f(x)=4x+58)A)B)C)D)Answer:CExplanation:A)B)C)D)7
Choose the function that might be used as a model for the data in the scatter plot.9)9)A)Polynomial, not quadraticB)Exponential, f(x)=ab-xor f(x)=P0e-kx, k>0C)Logistic, f(x)=a1+be-kxD)Logarithmic, f(x)=a+b ln xAnswer:DExplanation:A)B)C)D)8
10)10)A)Exponential, f(x)=abxor f(x)=P0ekx, k>0B)Logarithmic, f(x)=a+b ln xC)Polynomial, not quadraticD)Logistic, f(x)=a1+be-kxAnswer:DExplanation:A)B)C)D)9
Match the function with one of the graphs.11)f(x)=ex-311)A)B)C)D)Answer:BExplanation:A)B)C)D)10
Choose the function that might be used as a model for the data in the scatter plot.12)12)A)Polynomial, not quadraticB)Quadratic, f(x)=ax2+bx+cC)Exponential, f(x)=abxor f(x)=P0ekx, k>0D)Logarithmic, f(x)=a+b ln xAnswer:BExplanation:A)B)C)D)SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.For the function f, use composition of functions to show that f-1is as given.13)f(x)=x-26,f-1(x)=6x+213)Answer:Answers may vary. One possible solution is:1.(f-1Hf)(x)=f-1(f(x))=f-1x-2/6=6x-2/6+2=x-2+2=x;2. (fHf-1)(x)=f(f-1(x))=f6x+2=6x+2-2/6=6x/6=xExplanation:Provide an appropriate response.14)Explain how the equation log x=1 could be solved using the graph of f(x)=log x.14)Answer:Graph f(x)=log x and g(x)=1 on the same set of axes. The solution is the firstcoordinate of the point of intersection of the two graphs.Explanation:15)Prove that the function f is not one-to-one.f(x)=2-x415)Answer:Answers may vary. Possible answer:Find two numbers a and b for which aJb and f(a)=f(b). Two such numbers are-2and 2 because f(-2)=f(2)=-14. Thus, f is not one-to-one.Explanation:11
For the function f, use composition of functions to show that f-1is as given.16)f(x)=x3-7,f-1(x)=3x+716)Answer:Answers may vary. One possible solution is:1.(f-1Hf)(x)=f-1(f(x))=f-1×3-7=3×3-7+7=3×3=x;2. (fHf-1)(x)=f(f-1(x))=f3x+7=3x+73-7=x+7-7=xExplanation:Provide an appropriate response.17)Prove that the function f is one-to-one.f(x)=6-12×17)Answer:Assume that f(a)=f(b) for any numbers a and b in the domain of f.Since f(a)=6-12a and f(b)=6-12b, we have6-12a=6-12b-12a=-12ba=bThus if f(a)=f(b) then a=b and f is one-to-one.Explanation:18)Without using a calculator, explain why2Δmust be greater than 8 but less than 16.18)Answer:Since 3<Δ<4,23<2Δ<24; that is2Δis greater than23(or 8), but less than24(or16). This is true because2xis an increasing function.Explanation:19)Explain why 1 is excluded from being a logarithmic base.19)Answer:It excludes x=1ywhich is not a function.Explanation:20)The product, power, and quotient rules enable us to simplify expressions likelogarspv.Explain why such expressions can always be simplified without using the quotient rule.20)Answer:The quotientrspvcan be written as the product rs(p-1)(v-1)Explanation:21)Prove that the function f is not one-to-one.f(x)=x-221)Answer:Answers may vary. Possible answer:Find two numbers a and b for which aJb and f(a)=f(b). Two such numbers are-3and 3 because f(-3)=f(3)=1. Thus, f is not one-to-one.Explanation:12
22)Explain the error in the following:log68-log6N=log6(8-N).22)Answer:A difference of logarithms is not equal to the logarithm of the difference.log68-log6N=log68NExplanation:23)Explain whyaloga5=5.23)Answer:Letloga5=x. Thenax=5. Replacing x withloga5, we havealoga5=5.Explanation:24)Prove that the function f is not one-to-one.f(x)=x2+224)Answer:Answers may vary. Possible answer:Find two numbers a and b for which aJb and f(a)=f(b). Two such numbers are-1and 1 because f(-1)=f(1)=3. Thus, f is not one-to-one.Explanation:For the function f, use composition of functions to show that f-1is as given.25)f(x)=3x+5,f-1(x)=x3-525)Answer:1.(f-1Hf)(x)=f-1(f(x))=f-1(3x+5)=(3x+5)3-5=x;2. (fHf-1)(x)=f(f-1(x))=f(x3-5)=3(x3-5)+5=xExplanation:Provide an appropriate response.26)Explain why negative numbers do not have logarithms.26)Answer:Consider f(x)=logax, or y=logax, where a is a positive constant other than 1.Thenay=x, and since a is positive, x must be positive.Explanation:27)Explain why f(x)=2xis an exponential function but f(x)=x2is not.27)Answer:2xis an exponential function because the variable is in the exponent. In f(x)=x2, thevariable is not in the exponent; this is a polynomial function.Explanation:28)Without using a calculator, determine which of these numbers is larger:83or83.28)Answer:83=83/2and 3/2<3, so83is larger.Explanation:13
29)Suppose that $1000 is invested for 5 years at 4% interest, compounded annually. In whatyear will the most interest be earned? Why?29)Answer:The most interest will be earned in the fifth year, because the amount on which theinterest is paid is greatest at the end of the fourth year.Explanation:30)Find f(f-1(1247)) given that f(x)=55x-69x+4.30)Answer:1247Explanation:For the function f, use composition of functions to show that f-1is as given.31)f(x)=72x,f-1(x)=27×31)Answer:1.(f-1Hf)(x)=f-1(f(x))=f-172x=2772x=x;2. (fHf-1)(x)=f(f-1(x))=f27x=7227x=xExplanation:Provide an appropriate response.32)Without using a calculator, determine which of these numbers is larger:Δ1.3orΔ2.4.32)Answer:Since the bases are the same, the base with the larger exponent is the larger number.Thus,Δ2.4is larger.Explanation:33)Explain how the graph of f(x)=ln x could be used to graph the function g(x)=ex-1.33)Answer:Reflect the graph of f(x)=ln x across the line y=x and then translate it to the rightone unit.Explanation:34)Prove that the function f is not one-to-one.f(x)=2×634)Answer:Answers may vary. Possible answer:Find two numbers a and b for which aJb and f(a)=f(b). Two such numbers are-1and 1 because f(-1)=f(1)=2. Thus, f is not one-to-one.Explanation:For the function f, use composition of functions to show that f-1is as given.35)f(x)=x+49,f-1=9x-435)Answer:1.(f-1Hf)(x)=f-1(f(x))=f-1x+49=9x+49-4=x;2. (fHf-1)(x)=f(f-1(x))=f(9x-4)=9x-4+49=xExplanation:14
Provide an appropriate response.36)Prove that the function f is one-to-one.f(x)=x3-436)Answer:Assume that f(a)=f(b) for any numbers a and b in the domain of f.Since f(a)=a3-4 and f(b)=b3-4, we havea3-4=b3-4a3=b3a=bThus if f(a)=f(b) then a=b and f is one-to-one.Explanation:37)Explain the error in the following:log32+log3M=log3(2+M).37)Answer:A sum of logarithms is not equal to a logarithm of a sum.log32+log3M=log32MExplanation:38)Explain how the graph of f(x)=excould be used to graph the function g(x)=1+ln x.38)Answer:Reflect the graph of f(x)=exacross the line y=x and then translate it up one unit.Explanation:For the function f, use composition of functions to show that f-1is as given.39)f(x)=1+xx,f-1(x)=1x-139)Answer:Answers may vary. One possible solution is:1.(f-1Hf)(x)=f-1(f(x))=f-11+x/x=11+xx-1=11+x-xx=11x=x;2. (fHf-1)(x)=f(f-1(x))=f1x-1=1+1x-11x-1=1x-1+1x-11x-1=1×1=xExplanation:Provide an appropriate response.40)Explain whylog213 is between 3 and 4.40)Answer:log28=3 andlog216=4. Since 8<13<16, then 3<log213<4. This is truebecauselog2x is an increasing function.Explanation:15
41)Prove that the function f is one-to-one.f(x)=3x+541)Answer:Assume that f(a)=f(b) for any numbers a and b in the domain of f.Since f(a)=3a+5 and f(b)=3b+5, we have3a+5=3b+53a=3ba=bThus if f(a)=f(b) then a=b and f is one-to-one.Explanation:For the function f, use composition of functions to show that f-1is as given.42)f(x)=-67x,f-1(x)=-76×42)Answer:Answers may vary. One possible solution is:1.(f-1Hf)(x)=f-1(f(x))=f-1-67x=-76-67x=x;2. (fHf-1)(x)=f(f-1(x))=f-76x=-67-76x=xExplanation:Provide an appropriate response.43)Explain the error in the following:log43y=log43·log4y.43)Answer:A logarithm of a product is not equal to a product of logarithms.log43y=log43+log4yExplanation:44)Prove that the function f is one-to-one.f(x)=3x-144)Answer:Assume that f(a)=f(b) for any numbers a and b in the domain of f.Since f(a)=3a-1 and f(b)=3b-1, we have3a-1=3b-13a=3ba=bThus if f(a)=f(b) then a=b and f is one-to-one.Explanation:For the function f, use composition of functions to show that f-1is as given.45)f(x)=x+75,f-1(x)=5x-745)Answer:Answers may vary. One possible solution is:1.(f-1Hf)(x)=f-1(f(x))=f-1x+7/5=5x+7/5-7=x+7-7=x;2. (fHf-1)(x)=f(f-1(x))=f5x-7=5x-7+7/5=5x/5=xExplanation:16
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.Determine whether the function is one-to-one by graphing and using the horizontal line test.46)f(x)=9x+946)A)YesB)NoAnswer:AExplanation:A)B)Solve the logarithmic equation.47)log16x=1247)A)4B)256C)65,536D)0.00001526Answer:AExplanation:A)B)C)D)Find an equation of the inverse relation.48)y=-3×2-9x-248)A)y=-3×2+9x+2B)x=-3y2-9y-2C)x=-3y2+9y+2D)x=-3y2-9x-2Answer:BExplanation:A)B)x=-9±<b2-4ac><2a>C)D)17
Solve the problem.49)An element decays at the rate of S(t)=se-0.071t, where t is time in years and s is the initial size ingrams. If you have a42-gram piece of this element, how many grams will you have3years fromnow? Round your answer to the nearest tenth of a gram.49)A)0.8gB)52.0gC)33.9gD)19.6gAnswer:CExplanation:A)B)C)D)Evaluate to four decimal places using a calculator.50)e0.4450)A)1.1960B)1.5527C)1.7160D)-0.8210Answer:BExplanation:A)B)C)D)Graph the function. Describe its position relative to the graph of the indicated basic function.51)f(x)=e7x; relative to f(x)=ex51)A)Shrunk horizontallyB)Shrunk vertically18
C)Shrunk verticallyD)Shrunk horizontallyAnswer:DExplanation:A)B)C)D)Find the inverse of the function.52)f(x)=7×52)A)f-1(x)=x7B)f-1(x)=7+xC)f-1(x)=7xD)f-1(x)=7xAnswer:AExplanation:A)B)C)D)The graph of a one-to-one function f is given. Sketch the graph of the inverse functionf-1, on the same set of axes. Use adashed line for the inverse.53)53)19
A)B)C)D)Answer:CExplanation:A)B)C)D)Solve the problem.54)A bacterial culture has an initial population of 10,000. If its population declines to5000in6hours,when will its population be3969? Assume that the population decreases according to theexponential model.54)A)after 7 hrB)after 5 hrC)after8hrD)after 9 hrAnswer:CExplanation:A)B)C)D)20
Find the domain and range of the inverse of the given function.55)f(x)=x+1x-355)A)Domain and range: (-Q,Q)B)Domain: (-Q, 3)1(3,Q)Range: (-Q,1)1(1,Q)C)Domain: (-Q,Q)Range: (-Q, 3 )1(3,Q)D)Domain: (-Q,1)1(1,Q)Range: (-Q, 3)1(3,Q)Answer:DExplanation:A)B)C)D)Solve.56)The population of a particular city is increasing at a rate proportional to its size. It follows thefunction P(t)=1+ke0.05twhere k is a constant and t is the time in years. If the current populationis11,000, in how many years is the population expected to be27,500? (Round to the nearest year.)56)A)100yrB)8yrC)18yrD)10yrAnswer:CExplanation:A)B)C)D)Use the compound-interest formula to find the account balance with the given conditions:P=principal,r=interest rate,t=time, in years.57)P=$700, t=14, r=6.4% compounded daily57)A)$1710.77B)$1714.71C)$5,407,157.86D)$171.47Answer:BExplanation:A)B)C)D)Solve.58)There are currently54million cars in a certain country, decreasing by0.9% annually. How manyyears will it take for this country to have44million cars? Round to the nearest year.58)A)11yrB)5yrC)256yrD)23yrAnswer:DExplanation:A)B)C)D)Sketch the graph of the function. Describe how the graph can be obtained from the graph of a basic logarithmic function.21
59)f(x)=ln(x-5)59)A)Shift y=ln x right5unitsB)Shift y=ln x right5unitsC)Shift y=ln x left5unitsD)Shift y=ln x left5unitsAnswer:BExplanation:A)B)C)D)22
Determine whether the given function is one-to-one. If it is one-to-one, find a formula for the inverse.60)f(x)=3x-460)A)Not one-to-oneB)f-1(x)=3x+4C)f-1(x)=x3+4D)f-1(x)=x3-4Answer:CExplanation:A)B)C)D)Use the compound-interest formula to find the account balance with the given conditions:P=principal,r=interest rate,t=time, in years.61)P=$1000, t=5, r=7% compounded annually61)A)$1500.73B)$1310.80C)$402.55D)$1402.55Answer:DExplanation:A)B)C)D)Express as a sum of logarithms.62)logr8T62)A)log8+log TB)logr8+logrT+logrrC)logr8+logrTD)log7r+logTrAnswer:CExplanation:A)B)C)D)Graph the one-to-one function as a solid curve and its inverse as a dashed curve on the same axes.63)y=x+263)23
A)B)C)D)Answer:DExplanation:A)B)C)D)Evaluate to four decimal places using a calculator.64)e2.47764)A)7.3891B)6.1355C)6.7332D)11.9055Answer:DExplanation:A)B)C)D)24
Determine whether the function is one-to-one by graphing and using the horizontal line test.65)f(x)=64-x265)A)YesB)NoAnswer:BExplanation:A)B)Solve the exponential equation. Round to three decimal places when necessary.66)e-0.09t=0.0866)A)26.755B)-57.565C)-34.399D)28.064Answer:DExplanation:A)B)C)D)Graph the function.67)f(x)=2(x+1)67)25
A)B)C)D)Answer:DExplanation:A)B)C)D)Use the compound-interest formula to find the account balance with the given conditions:P=principal,r=interest rate,t=time, in years.68)P=$1000, t=11, r=13% compounded semiannually68)A)$3996.61B)$2996.61C)$3752.68D)$3835.86Answer:AExplanation:A)B)C)D)Graph the equation as a solid line. Graph the inverse relation as a dashed line on the same axes by reflecting across theline y=x.26
69)y=52x+269)A)B)C)D)Answer:AExplanation:A)B)C)D)27
Determine whether the given function is one-to-one. If it is one-to-one, find a formula for the inverse.70)f(x)=x-10x-670)A)f-1(x)=-6x-10x-1B)f-1(x)=6x-1x-10C)f-1(x)=6x-10x-1D)f-1(x)=x+10-6+xAnswer:CExplanation:A)B)C)D)Solve.71)The sales of typewriters have declined considerably in recent years because of the advent ofcomputers. Assume that there were24million typewriters sold in 1980 and4million sold in 1990.In addition, assume the sales are decreasing according to the exponential decay model. Find thevalue k, and write an exponential function that describes the number of typewriters sold after timet, in years. (Round decimals to three places.)71)A)k=0.139; S(t)=24e-0.139t, where S(t) is in millions and t is the number of years since 1980.B)k=0.179; S(t)=24e-0.179t, where S(t) is in millions and t is the number of years since 1980.C)k=0.318; S(t)=24e-0.318t, where S(t) is in millions and t is the number of years since 1980.D)k=-0.179; S(t)=24e-0.179t, where S(t) is in millions and t is the number of years since 1980.Answer:BExplanation:A)B)C)D)72)Given thatloga2=0.3010 andloga3=0.4771, findloga72.72)A)1.8572B)2.0333C)0.2549D)0.8616Answer:AExplanation:A)B)C)D)Solve the exponential equation.73)62x=6973)A)-4.5B)27C)6D)4.5Answer:DExplanation:A)B)C)D)28
Find an equation of the inverse relation.74)x=y-6y274)A)y=x+6y2B)y=x-6x2C)y=x-6y2D)y=x+6x2Answer:BExplanation:A)B)C)D)Express as a single logarithm and, if possible, simplify.75)logb8x+2(logbx-logby)75)A)logb8x3y2B)logb16x2y2C)(logb8x)(logbx2)logby2D)logb(8x+x2-y2)Answer:AExplanation:A)B)C)D)Find the value of the expression.76)log8876)A)0B)1C)8D)10Answer:BExplanation:A)B)C)D)Graph the piecewise function.77)f(x)=x+1,ln(x-1),for xK1,for x>177)29
A)B)C)D)Answer:BExplanation:A)B)C)D)Find the inverse of the function.78)f(x)=3x-278)A)f-1(x)=x3+2B)f-1(x)=(x+2)3C)f-1(x)=3x-2D)f-1(x)=(x-2)3Answer:AExplanation:A)B)C)D)Sketch the graph of the function. Describe how the graph can be obtained from the graph of a basic logarithmic function.30
79)f(x)=log2(x-2)79)A)Shift y=log2x left2unitsB)Shift y=log2x right2unitsC)Shift y=log2x left2unitsD)Shift y=log2x right2unitsAnswer:BExplanation:A)B)C)D)31
Find the logarithm using the change-of-base formula.80)log3003080)A)0.5963B)5.7038C)1.677D)3.4012Answer:AExplanation:A)B)C)D)Sketch the graph of the function. Describe how the graph can be obtained from the graph of a basic exponential function.81)f(x)=3(x+2)81)A)Shift y=3xdown2unit(s)B)Shift y=3xup2unit(s)C)Shift y=3xleft2unit(s)D)Shift y=3xright2unit(s)Answer:CExplanation:A)B)C)32
C)D)Solve.82)If the population of a certain country doubles in15.40years, find the growth rate k. Assume that thepopulation increases exponentially.82)A)5.7%B)4.5%C)8.2%D)3.1%Answer:BExplanation:A)B)C)D)Graph the function and its inverse using the same set of axes. Use any method.83)f(x)=5x;f-1(x)=log5x83)A)B)33
C)D)Answer:CExplanation:A)B)C)D)Find the value of the expression.84)log55784)A)7B)7log55C)1D)57Answer:AExplanation:A)B)C)D)Find the following using a calculator. Round to four decimal places.85)log1085)A)1B)0.5C)2.8026D)2.3026Answer:AExplanation:A)B)C)D)Graph the function.34
86)f(x)=13x+486)A)B)C)D)Answer:BExplanation:A)B)C)D)35
Solve the logarithmic equation.87)log3x=log4+log (x-2)87)A)1B)-8C)-87D)8Answer:DExplanation:A)B)C)D)Using the horizontal-line test, determine whether the function is one-to-one.88)f(x)=8×2-388)A)YesB)NoAnswer:BExplanation:A)B)Find the logarithm using the change-of-base formula.89)log8.27.189)A)0.8513B)0.8659C)1.0735D)0.9315Answer:DExplanation:A)B)C)D)Solve the logarithmic equation.90)log7x=-290)A)149B)49C)128D)1128Answer:AExplanation:A)B)C)D)The graph of a one-to-one function f is given. Sketch the graph of the inverse functionf-1, on the same set of axes. Use adashed line for the inverse.36
91)91)A)B)C)D)Answer:BExplanation:A)B)C)D)37
Simplify.92)10log t92)A)1B)10tC)10tD)tAnswer:DExplanation:A)B)C)D)Graph the one-to-one function as a solid curve and its inverse as a dashed curve on the same axes.93)f(x)=5×93)A)B)C)D)Answer:BExplanation:A)B)C)38
C)D)Find the logarithm using the change-of-base formula.94)log4294)A)-0.5000B)2.0000C)1.0000D)0.5000Answer:DExplanation:A)B)C)D)Graph the function.95)x=12y95)A)B)C)D)39
Answer:DExplanation:A)B)C)D)Graph the equation as a solid line. Graph the inverse relation as a dashed line on the same axes by reflecting across theline y=x.96)x=4y96)A)B)C)D)Answer:DExplanation:A)B)C)D)40
Find the inverse of the function.97)f(x)=-7×97)A)f-1(x)=-x7B)f-1(x)=7xC)f-1(x)=-7xD)f-1(x)=-7xAnswer:CExplanation:A)B)C)D)The graph of a one-to-one function f is given. Sketch the graph of the inverse functionf-1, on the same set of axes. Use adashed line for the inverse.98)98)A)B)C)D)41
Answer:DExplanation:A)B)C)D)Find the inverse of the relation.99){(-7,-5), (7,5), (-6,7), (6,-7)}99)A){(-5,-7), (5,7), (7,7), (-7,6)}B){(-5,-7), (-7,7), (7,-6), (-7,6)}C){(5,-7), (8,7), (7,-6), (-7,-6)}D){(-5,-7), (5,7), (7,-6), (-7,6)}Answer:DExplanation:A)B)C)D)Use the compound-interest formula to find the account balance with the given conditions:P=principal,r=interest rate,t=time, in years.100)P=$1700, t=6, r=5% compounded daily100)A)$34,075.39B)$229.47C)$2294.71D)$2293.33Answer:CExplanation:A)B)C)D)Find an equation of the inverse relation.101)x3y=-6101)A)x3y=-6B)x3=y-6C)y3=-6D)y3x=-6Answer:DExplanation:A)B)C)D)The graph of a one-to-one function f is given. Sketch the graph of the inverse functionf-1, on the same set of axes. Use adashed line for the inverse.42
102)102)A)B)C)D)Answer:DExplanation:A)B)C)D)43
Find the domain and range of the inverse of the given function.103)f(x)=3-x2; xL0103)A)Domain: [0,Q); range: (-Q,3]B)Domain: [3,Q); range: [0,Q)C)Domain: (-Q,3]; range: [0,Q)D)Domain: (-Q,3]; range: all real numbersAnswer:CExplanation:A)B)C)D)Find the domain and the vertical asymptote of the function.104)f(x)=6-log5x104)A)Domain: (0,Q); vertical asymptote: x=0B)Domain: [0,Q); vertical asymptote: noneC)Domain: [0,Q); vertical asymptote: x=0D)Domain: (6,Q); vertical asymptote: x=6Answer:AExplanation:A)B)C)D)Express in terms of sums and differences of logarithms.105)log157mn105)A)log157+12log15m-log15nB)log15n-log157-12log15mC)log15(7m)-log15nD)log157·12log15m÷log15nAnswer:AExplanation:A)B)C)D)Express as a sum of logarithms.106)log210x106)A)log210-log2xB)log210+log2xC)log110+log1xD)log110-log1xAnswer:BExplanation:A)B)C)D)44
Find the domain and range of the inverse of the given function.107)f(x)=x-5107)A)Domain: all real numbers; range: [5,Q)B)Domain and range: all real numbersC)Domain: [5,Q); range: [0,Q)D)Domain: [0,Q); range: [5,Q)Answer:DExplanation:A)B)C)D)Solve the logarithmic equation.108)log5(x+3)+log5(x-3)=3108)A)1253B)134C)134D)485Answer:BExplanation:A)B)C)D)Solve the problem.109)An artifact is discovered at a certain site. If it has58% of the carbon-14 it originally contained,what is the approximate age of the artifact? (carbon-14 decays at the rate of 0.012% annually.)109)A)3500yrB)4539yrC)1971yrD)4833yrAnswer:BExplanation:A)B)C)D)110)In September 1998 the population of the country of West Goma in millions was modeled byf(x)=17.4e0.0008x.At the same time the population of East Goma in millions was modeled byg(x)=13.6e0.0178x.In both formulas x is the year, wherex=0corresponds to September 1998.Assuming these trends continue, estimate the year when the population of West Goma will equalthe population of East Goma.110)A)1984B)2012C)14D)2011Answer:BExplanation:A)B)C)D)Sketch the graph of the function. Describe how the graph can be obtained from the graph of a basic exponential function.45
111)f(x)=2(x-3)111)A)Shift y=2xup3unit(s)B)Shift y=2xleft3unit(s)C)Shift y=2xright3unit(s)D)Shift y=2xdown3unit(s)Answer:CExplanation:A)B)C)D)46
Solve the problem.112)In September 1998 the population of the country of West Goma in millions was modeled byf(x)=17.2e0.0015x.At the same time the population of East Goma in millions was modeled byg(x)=14.6e0.0174x.In both formulas x is the year, wherex=0corresponds to September 1998.Assuming these trends continue, estimate what the population will be when the populations areequal.112)A)15millionB)17millionC)16millionD)1millionAnswer:BExplanation:A)B)C)D)Sketch the graph of the function. Describe how the graph can be obtained from the graph of a basic exponential function.113)f(x)=-2(x+3)113)A)Shift y=2xright3unit(s) andreflect across the x-axisB)Shift y=2xright3unit(s) andreflect across the x-axis47
C)Shift y=2xleft3unit(s) andreflect across the x-axisD)Shift y=2xleft3unit(s) andreflect across the x-axisAnswer:CExplanation:A)B)C)D)Express as a product.114)log10x10114)A)10log10xB)log1010C)log1010xD)log1010·log10xAnswer:AExplanation:A)B)C)D)Graph the function.115)f(x)=log x115)48
A)B)C)D)Answer:BExplanation:A)B)C)D)Simplify.116)7log7(5x)116)A)1B)5xC)75xD)7Answer:BExplanation:A)B)C)D)Sketch the graph of the function. Describe how the graph can be obtained from the graph of a basic exponential function.49
117)f(x)=2(x+1)-4117)A)Shift y=2xright1unit(s) andup4unit(s)B)Shift y=2xleft1unit(s) anddown4unit(s)C)Shift y=2xright1unit(s) anddown4unit(s)D)Shift y=2xleft1unit(s) andup4unit(s)Answer:BExplanation:A)B)C)D)50
Using the horizontal-line test, determine whether the function is one-to-one.118)1.8×118)A)YesB)NoAnswer:AExplanation:A)B)Evaluate to four decimal places using a calculator.119)e2.4119)A)6.5239B)0.8755C)12.1825D)11.0232Answer:DExplanation:A)B)C)D)Find the value of the expression.120)log1111120)A)-2B)-12C)12D)2Answer:CExplanation:A)B)C)D)Find the inverse of the relation.121){(6,4), (11,5), (9,6), (7,7)}121)A){(6,4), (5,11), (9,6), (7,7)}B){(5,4), (7,9), (6,9), (6,6)}C){(4,6), (5,11), (6,9), (7,7)}D){(5,4), (4,9), (6,11), (6,6)}Answer:CExplanation:A)B)C)D)51
Express in terms of sums and differences of logarithms.122)lnx6y122)A)6ln x+ln yB)ln3x+12yC)6ln x+ln yD)3ln x+12ln yAnswer:DExplanation:A)B)C)D)Find the domain and range of the inverse of the given function.123)f(x)=34x+13x+8123)A)Domain:gQ,3431343,QRange:gQ,83183,QB)Domain:gQ,336313363,QRange:gQ,-831-83,QC)Domain:gQ,3961396,QRange:gQ,-86186,QD)Domain:gQ,3361336,QRange:gQ,-831-83,QAnswer:BExplanation:A)B)C)D)Solve.124)Given thatloga8=0.903, andloga2=0.301, findloga64.124)A)1.806B)0.272C)1.204D)0.815Answer:AExplanation:A)B)C)D)52
Determine whether the given function is one-to-one. If it is one-to-one, find a formula for the inverse.125)f(x)=-9x+6-6x-5125)A)Not one-to-oneB)f-1(x)=-9x+6-6x-5C)f-1(x)=5x+6-6x+9D)f-1(x)=-6x+95x+6Answer:CExplanation:A)B)C)D)Express in terms of sums and differences of logarithms.126)logb4x8b7y4z10126)A)2logbx+74-logby+52logbzB)2logbx·74-(logby·52logbz)C)2logbx+7-4logby-10logbzD)2logbx+74-logby-52logbzAnswer:DExplanation:A)B)C)D)Find the value of the expression.127)lne4127)A)1B)14C)4D)e ln4Answer:CExplanation:A)B)C)D)Solve.128)How long will it take for $2400to grow to $21,900at an interest rate of4.7% if the interest iscompounded quarterly? Round the number of years to the nearest hundredth.128)A)189.28yrB)48.14yrC)195.29yrD)47.32yrAnswer:DExplanation:A)B)C)D)53
Solve the problem.129)Suppose the amount of a radioactive element remaining in a sample of 100 milligrams after x yearscan be described byA(x)=100e-0.02279x. How much is remaining after164years? Round theanswer to the nearest hundredth of a milligram.129)A)0.02mgB)373.76mgC)4199.54mgD)2.38mgAnswer:DExplanation:A)B)C)D)Graph the function.130)f(x)=e1-x130)A)B)C)D)Answer:AExplanation:A)B)54
B)C)D)Solve the exponential equation.131)23x=10-5×131)A)23B)0C)115D)-5Answer:BExplanation:A)B)C)D)Find the domain and range of the inverse of the given function.132)f(x)=3x-4.9132)A)Domain and range: all real numbersB)Domain: all real numbers; range: [0,Q)C)Domain: [4.9,Q); range: [0,Q)D)Domain:all real numbers; range:[4.9,Q)Answer:AExplanation:A)B)C)D)Find the logarithm using the change-of-base formula.133)log1587.07133)A)0.6063B)1.9399C)1.6494D)5.8047Answer:CExplanation:A)B)C)D)Graph the equation as a solid line. Graph the inverse relation as a dashed line on the same axes by reflecting across theline y=x.134)x=y2-2134)55
A)B)C)D)Answer:AExplanation:A)B)C)D)Find the domain and the vertical asymptote of the function.135)g(x)=ln(x-4)135)A)Domain:(-4,Q); vertical asymptote: x=-4B)Domain:(4,Q); vertical asymptote: x=4C)Domain: (-Q,Q); vertical asymptote: noneD)Domain:(0,Q); vertical asymptote: x=0Answer:BExplanation:A)B)C)D)56
Solve the logarithmic equation.136)ln x=7136)A)e7B)10,000,000C)ln7D)7eAnswer:AExplanation:A)B)C)D)Solve.137)Suppose that $5000is invested at an interest rate of5.1% per year, compounded continuously. Whatis the doubling time?137)A)2 yrB)14.6yrC)13.6yrD)12.6yrAnswer:CExplanation:A)B)C)D)Graph the function. Describe its position relative to the graph of the indicated basic function.138)f(x)=e-0.1x; relative to f(x)=ex138)A)Stretched verticallyB)Stretched horizontally;reflected across the y-axis57
C)Stretched vertically;reflected across the y-axisD)Stretched horizontallyAnswer:BExplanation:A)B)C)D)Sketch the graph of the function. Describe how the graph can be obtained from the graph of a basic exponential function.139)f(x)=-3(x-4)139)A)Reflect y=3xacross the x-axis andshift down4unit(s)B)Shift y=3xleft4unit(s) and reflectacross the x-axis58
C)Reflect y=3xacross the x-axis andshift up4unit(s)D)Shift y=3xright4unit(s) and reflectacross the x-axisAnswer:DExplanation:A)B)C)D)Determine whether the function is one-to-one by graphing and using the horizontal line test.140)f(x)=2.35140)A)YesB)NoAnswer:BExplanation:A)B)Graph the function.59
141)f(x)=3e-x141)A)B)C)D)Answer:BExplanation:A)B)C)D)60
Solve the problem.142)Newton’s Law of Cooling states that if a body with temperatureT1is placed in surroundings withtemperatureT0different fromT1, then the body will either cool or warm to temperature T(t) aftertime t, in minutes, whereT(t)=T0+(T1-T0)e-kt.A cup of coffee with temperature103°F is placed in a freezer with temperature 0°F. After8minutes, the temperature of the coffee is55.6°F. What will its temperature be16minutes after it isplaced in the freezer? Round your answer to the nearest degree.142)A)23°FB)28°FC)30°FD)25°FAnswer:CExplanation:A)B)C)D)Graph the function. Describe its position relative to the graph of the indicated basic function.143)f(x)=2(2-ex), x>0; relative to f(x)=ex143)A)Stretched horiz; reflected across y-axis;moved down2unit(s)B)Stretched vert; reflected across y-axis;moved up2unit(s)61
C)Stretched vert; reflected across x-axis;moved up4unit(s)D)Stretched horiz; reflected across x-axis;moved down4unit(s)Answer:CExplanation:A)B)C)D)Graph the function and its inverse using the same set of axes. Use any method.144)f(x)=-3log x;f-1(x)=10x/-3144)A)B)62
C)D)Answer:BExplanation:A)B)C)D)Solve.145)Find the hydrogen ion concentration of a solution whose pH is7.3. Use the formulapH=-log [H+].145)A)-0.8633229B)2.00×107C)0.86332286D)5.01×10-8Answer:DExplanation:A)B)C)D)Graph the piecewise function.146)f(x)=1,log x-1,for xK0,for x>0146)63
A)B)C)D)Answer:AExplanation:A)B)C)D)Sketch the graph of the function. Describe how the graph can be obtained from the graph of a basic exponential function.147)f(x)=3(x-3)-2147)64
A)Shift y=3xright3unit(s) andup2unit(s)B)Shift y=3xleft3unit(s) andup2unit(s)C)Shift y=3xleft3unit(s) anddown2unit(s)D)Shift y=3xright3unit(s) anddown2unit(s)Answer:DExplanation:A)B)C)D)Find the value of the expression.148)log10100148)A)200B)100C)2D)-2Answer:CExplanation:A)B)C)D)65
Use the compound-interest formula to find the account balance with the given conditions:P=principal,r=interest rate,t=time, in years.149)P=$12,000, t=8, r=12% compounded quarterly149)A)$30,900.99B)$29,711.56C)$30,000.96D)$18,900.99Answer:AExplanation:A)B)C)D)Express in terms of sums and differences of logarithms.150)logbm2p3n5b9150)A)logb (m2p3-n5b9)B)2logbm·3logbp÷5logbn-9C)2logbm+3logbp-5logbn+9D)2logbm+3logbp-5logbn-9Answer:DExplanation:A)B)C)D)Find the logarithm using the change-of-base formula.151)log20040151)A)1.4363B)5.2983C)3.6889D)0.6962Answer:DExplanation:A)B)C)D)66
Using the horizontal-line test, determine whether the function is one-to-one.152)f(x)=3x+6-6152)A)YesB)NoAnswer:AExplanation:A)B)Convert to a logarithmic equation.153)e-7=0.0009119153)A)0.0009119=log-7eB)0.0009119=loge-7C)e=log-70.0009119D)-7=loge0.0009119Answer:DExplanation:A)B)C)D)Solve the exponential equation. Round to three decimal places when necessary.154)ex-24e-x=2154)A)1.668B)1.915C)1.804D)1.792Answer:DExplanation:A)B)C)D)Graph the function.67
155)x=5y155)A)B)C)D)Answer:AExplanation:A)B)C)D)Sketch the graph of the function. Describe how the graph can be obtained from the graph of a basic exponential function.68
156)f(x)=2x+4156)A)Shift y=2xdown4unit(s)B)Shift y=2xdown4unit(s)C)Shift y=2xup4unit(s)D)Shift y=2xup4unit(s)Answer:DExplanation:A)B)C)D)69
Simplify.157)6log6(2x-2)157)A)6(2x-2)B)1C)2x-2D)6Answer:CExplanation:A)B)C)D)Solve the exponential equation. Round to three decimal places when necessary.158)8x-8-x8x+8-x=10158)A)No solutionB)0C)2.303D)0.226Answer:AExplanation:A)B)C)D)Express as a sum of logarithms.159)log60.10y159)A)log60.10+yB)log60.10+log6yC)log 0.10+log yD)log6+log 0.10Answer:BExplanation:A)B)C)D)Convert to a logarithmic equation.160)3-2=19160)A)-2=log319B)-2=log1/93C)19=log3-2D)3=log219Answer:AExplanation:A)B)C)D)Graph the function.70
161)f(x)=ex-4161)A)B)C)D)Answer:CExplanation:A)B)C)D)Graph the equation as a solid line. Graph the inverse relation as a dashed line on the same axes by reflecting across theline y=x.71
162)y=5×162)A)B)C)D)Answer:AExplanation:A)B)C)D)72
Solve.163)Given thatloga3=1.099, andloga11=2.398, findloga311.163)A)0.458B)3.497C)-1.299D)1.299Answer:CExplanation:A)B)C)D)Graph the function.164)y=log3x164)A)B)C)D)Answer:AExplanation:A)B)73
B)C)D)Express in terms of sums and differences of logarithms.165)logax2yz5165)A)(logax)2+logay+(logaz)5B)2 logax+logay+5logazC)2 logax·logay·5logazD)loga2x+logay+loga5zAnswer:BExplanation:A)B)C)D)Sketch the graph of the function. Describe how the graph can be obtained from the graph of a basic exponential function.166)f(x)=2·3(x-4)+4166)A)Shift y=3xright4unit(s), stretch itvertically, and shift it up4unit(s)B)Shift y=3xleft4unit(s), stretch itvertically, and shift it up4unit(s)74
C)Shift y=3xright4unit(s), stretch itvertically, reflect across the x-axis, andshift it up4unit(s)D)Shift y=3xleft4unit(s), stretch itvertically, reflect across the x-axis andshift it up4unit(s)Answer:AExplanation:A)B)C)D)Find the following using a calculator. Round to four decimal places.167)ln0.999167)A)-0.0010B)0.0004C)0.0010D)-0.0004Answer:AExplanation:A)B)C)D)Solve the exponential equation.168)2(1+2x)=8168)A)4B)-1C)2D)1Answer:DExplanation:A)B)C)D)Convert to a logarithmic equation.169)52=25169)A)2=log255B)25=log52C)5=log225D)2=log525Answer:DExplanation:A)B)C)D)75
Solve the exponential equation. Round to three decimal places when necessary.170)e-t=0.03170)A)3.607B)-3.507C)3.307D)3.507Answer:DExplanation:A)B)C)D)Express as a single logarithm and, if possible, simplify.171)12log s+2logt171)A)logst22B)logt2sC)logs2+2tD)log (t2+s)Answer:BExplanation:A)B)C)D)Find the inverse of the relation.172){(4,8), (5,8), (6,-1), (7,1)}172)A){(8,4), (8,5), (-1,6), (1,7)}B){(4,1), (4,-1), (6,5)}C){(8,4), (5,-1), (6,1)}D){(8,4), (1,-1), (6,5)}Answer:AExplanation:A)B)C)D)Simplify.173)elnx6173)A)lnx6B)xC)ex6D)x6Answer:DExplanation:A)B)C)D)Solve.174)The population of a small country increases according to the functionB=1,200,000e0.02t,where t ismeasured in years. How many people will the country have after2years?174)A)3,862,651B)1,315,774C)1,248,973D)1,677,528Answer:CExplanation:A)B)C)D)Sketch the graph of the function. Describe how the graph can be obtained from the graph of a basic logarithmic function.76
175)f(x)=6ln x175)A)Stretch y=ln x verticallyB)Shift y=ln x up6unitsC)Stretch y=ln x horizontallyD)Shrink y=ln x verticallyAnswer:AExplanation:A)B)C)D)77
Find the domain and the vertical asymptote of the function.176)f(x)=5+log3x176)A)Domain: (0,Q); vertical asymptote: noneB)Domain: (0,Q); vertical asymptote: x=0C)Domain: (5,Q); vertical asymptote: x=5D)Domain: [0,Q); vertical asymptote: x=0Answer:BExplanation:A)B)C)D)Convert to an exponential equation.177)ln41=3.7136177)A)e3.7136=41B)e3.7136=ln41C)e41=3.7136D)e3.7136=1Answer:AExplanation:A)B)C)D)Express as a single logarithm and, if possible, simplify.178)ln (x2-4)-ln (x+2)178)A)ln (x-4)B)ln (x2-2)C)ln (x-2)D)ln (x+2)Answer:CExplanation:A)B)C)D)Solve the logarithmic equation.179)ln(4x-5)=ln5-ln (x-1)179)A)1,14B)94C)+D)0,94Answer:BExplanation:A)B)C)D)Convert to an exponential equation.180)ln8a=2180)A)e2=8aB)e8=2aC)ea=28D)2e=8aAnswer:AExplanation:A)B)C)D)78
Solve.181)In 1985, the number of female athletes participating in Summer Olympic-Type Games was550. In1996, about3550participated in the Summer Olympics in Atlanta. Assuming thatP(0)=500andthat the exponential model applies, find the value of k rounded to the hundredths place, and writethe function.181)A)k=0.17; P(t)=500e0.27tB)k=0.19; P(t)=500e0.19tC)k=0.17; P(t)=500e0.17tD)k=0.16; P(t)=500e0.16tAnswer:CExplanation:A)B)C)D)Determine whether the given function is one-to-one. If it is one-to-one, find a formula for the inverse.182)f(x)=6×2+1, xL0182)A)f-1(x)=x-16B)f-1(x)=6x-1C)f-1(x)=6x-1D)Not one-to-oneAnswer:AExplanation:A)B)C)D)Express as a product.183)logbL-8183)A)logbL·logb-8B)-8logbLC)logbL·logbLD)Llogb-8Answer:BExplanation:A)B)C)D)Simplify.184)logeex-11184)A)loge11B)11logeeC)x-11D)logx-11Answer:CExplanation:A)B)C)D)79
Express in terms of sums and differences of logarithms.185)logbx5y2z8185)A)52logbx·logby÷4logbzB)52logbx-logby+4logbzC)52logbx+logby-4logbzD)52logbx+2logby-8logbzAnswer:CExplanation:A)B)C)D)Convert to an exponential equation.186)log2116=-4186)A)42=116B)2-4=116C)1164=2D)216=4Answer:BExplanation:A)B)C)D)Find the following using a calculator. Round to four decimal places.187)ln21,200,000187)A)5.3566B)0.0591C)7.3263D)16.8695Answer:DExplanation:A)B)C)D)Express as a single logarithm and, if possible, simplify.188)ln27-ln3188)A)ln3B)ln(24)C)ln13D)ln(9)Answer:DExplanation:A)B)C)D)80
Find a formula for the inverse of the function described below.189)An organization determines that the cost per person of chartering a bus is given by the formulaC(x)=150+7xx,where x is the number of people in the group and C(x) is in dollars.189)A)C-1(x)=7x-150B)C-1(x)=150+x7C)C-1(x)=150x+7D)C-1(x)=150x-7Answer:DExplanation:A)B)C)D)Find the value of the expression.190)log216190)A)4B)8C)16D)2Answer:AExplanation:A)B)C)D)Express as a single logarithm and, if possible, simplify.191)2logcm-logcn191)A)logcm2nB)logc2mnC)logcm2÷logcnD)logc(m2-n)Answer:AExplanation:A)B)C)D)Solve the logarithmic equation.192)log4(x-4)+log4(x-4)=1192)A)-17,17B)6C)-6,6D)17Answer:BExplanation:A)B)C)D)81
Find an equation of the inverse relation.193)y=-4x-2193)A)x=-4y-2B)y=-4-2xC)x=-4y+2D)y=-4x+2Answer:AExplanation:A)B)C)D)Graph the piecewise function.194)f(x)=-3,for xK-2,×2+4,for-2<xK0,ex,for x>0194)A)B)82
C)D)Answer:DExplanation:A)B)C)D)Solve.195)Given thatloga2=0.3010 andloga3=0.4771, findloga98195)A)0.8293B)0.0512C)2.0333D)0.1992Answer:BExplanation:A)B)C)D)Find the following using a calculator. Round to four decimal places.196)log93,900196)A)6.8449B)6.845C)4.9727D)6.8447Answer:CExplanation:A)B)C)D)Graph the function and its inverse using the same set of axes. Use any method.83
197)f(x)=ex;f-1(x)=ln x197)A)B)C)D)Answer:DExplanation:A)B)C)D)84
Find the domain and the vertical asymptote of the function.198)f(x)=-5-4log(x+2)198)A)Domain (-Q,2); vertical asymptote: x=2B)Domain (-Q,-2); vertical asymptote: x=-2C)Domain (2,Q); vertical asymptote: x=2D)Domain (-2,Q); vertical asymptote: x=-2Answer:DExplanation:A)B)C)D)Use the compound-interest formula to find the account balance with the given conditions:P=principal,r=interest rate,t=time, in years.199)P=$4219.14, t=11, r=6% compounded monthly199)A)$8149.75B)$8084.31C)$8009.19D)$8123.25Answer:AExplanation:A)B)C)D)Solve the problem.200)There are currently54million cars in a certain country, decreasing exponentially by6.8%annually. How many years will it take for this country to have46million cars? Round to thenearest year.200)A)8yrB)1yrC)31yrD)2yrAnswer:DExplanation:A)B)C)D)Find the following using a calculator. Round to four decimal places.201)log0.03714201)A)-2.7931B)-3.2931C)-1.4302D)-1.9302Answer:CExplanation:A)B)C)D)85
Find the inverse of the function.202)f(x)=3-x202)A)f-1(x)=3-xB)f-1(x)=x+3C)f-1(x)=x3D)f-1(x)=x-3Answer:AExplanation:A)B)C)D)Solve.203)A lake is stocked with694fish of a new variety. The size of the lake, the availability of food, and thenumber of other fish restrict growth in the lake to a limiting value of4337. The population of fish inthe lake after time t, in months, is given by the function P(t)=43371+5.74e-0.34t. Find the populationafter1month(s).203)A)868B)853C)863D)843Answer:BExplanation:A)B)C)D)Express as a single logarithm and, if possible, simplify.204)loga3x-loga3x204)A)loga3xB)loga1xC)loga3x-3xD)loga3x6Answer:AExplanation:A)B)C)D)Convert to an exponential equation.205)log4T9=6205)A)64=T9B)46=T9C)6T9=4D)4T9=6Answer:BExplanation:A)B)C)D)86
Solve the exponential equation. Round to three decimal places when necessary.206)e-x=32×206)A)No solutionB)3C)0D)0.549Answer:CExplanation:A)B)C)D)Determine whether the function is one-to-one by graphing and using the horizontal line test.207)f(x)=3x-3207)A)YesB)NoAnswer:AExplanation:A)B)Graph the function.208)f(x)=e3x-4208)87
A)B)C)D)Answer:CExplanation:A)B)C)D)Solve the exponential equation.209)4x=64209)A)2B)4C)16D)3Answer:DExplanation:A)B)C)D)Find the domain and range of the inverse of the given function.210)f(x)=x3-5210)A)Domain: [-5,Q); range: all real numbersB)Domain and range: all real numbersC)Domain: all real numbers; range: [-5,Q)D)Domain: [0,Q); range: [0,Q)Answer:BExplanation:A)B)C)D)Sketch the graph of the function. Describe how the graph can be obtained from the graph of a basic logarithmic function.88
211)f(x)=ln x-4211)A)Shift y=ln x down4unitsB)Shift y=ln x right4unitsC)Shift y=ln x up4unitsD)Shift y=ln x down4unitsAnswer:DExplanation:A)B)C)D)Graph the function.89
212)f(x)=ln x212)A)B)C)D)Answer:DExplanation:A)B)C)D)Graph the function and its inverse using the same set of axes. Use any method.90
213)f(x)=log x;f-1(x)=10×213)A)B)C)D)Answer:BExplanation:A)B)C)D)91
Solve the problem.214)The number of reports of a certain virus has increased exponentially since 1960. The currentnumber of cases can be approximated using the functionr(t)=195e0.007t,where t is the number ofyears since 1960. Estimate the of cases in the year 2000.214)A)194casesB)241casesC)516casesD)258casesAnswer:DExplanation:A)B)C)D)Solve.215)There are currently69million cars in a certain country, increasing exponentially by5.1% annually.How many years will it take for this country to have89million cars? Round to the nearest year.215)A)3yrB)4yrC)59yrD)5yrAnswer:DExplanation:A)B)C)D)Determine whether the function is one-to-one by graphing and using the horizontal line test.216)f(x)=-x3-8216)A)YesB)NoAnswer:AExplanation:A)B)Find the inverse of the relation.217){(6,-15), (20,8), (15,-17)}217)A){(6,8), (6,20), (-17,15)}B){(-15,6), (8,20), (-17,15)}C){(6,-15), (20,8), (15,-17)}D){(-15,6), (15,20), (-17,8)}Answer:BExplanation:A)B)C)D)92
Express as a difference of logarithms.218)loggM60218)A)loggM÷logg60B)loggM-logg60C)logg60-loggMD)logg(M-60)Answer:BExplanation:A)B)C)D)Determine whether the function is one-to-one by graphing and using the horizontal line test.219)f(x)=x-3219)A)YesB)NoAnswer:BExplanation:A)B)Sketch the graph of the function. Describe how the graph can be obtained from the graph of a basic logarithmic function.220)f(x)=5-3log(x-5)220)93
A)Shift y=log x left5units, stretch itvertically, reflect it over the x-axis, andshift up5unitsB)Shift y=log x left5units, stretch itvertically, and shift up5unitsC)Shift y=log x left5units, stretch itvertically, and shift down5unitsD)Shift y=log x right5units, stretch itvertically, reflect it over the x-axis, andshift up5unitsAnswer:DExplanation:A)B)C)D)Find the value of the expression.221)log21128221)A)1/7B)14C)-7D)2Answer:CExplanation:A)B)C)D)94
Determine whether the given function is one-to-one. If it is one-to-one, find a formula for the inverse.222)f(x)=3×3-2222)A)f-1(x)=3x-23B)Not one-to-oneC)f-1(x)=3x+23D)f-1(x)=3×3+2Answer:CExplanation:A)B)C)D)Sketch the graph of the function. Describe how the graph can be obtained from the graph of a basic exponential function.223)f(x)=4x-3-1223)A)Shift y=4xleft3unit(s) anddown1unit(s)B)Shift y=4xright3unit(s) andup1unit(s)95
C)Shift y=4xright3unit(s) anddown1unit(s)D)Shift y=4xleft3unit(s) andup1unit(s)Answer:CExplanation:A)B)C)D)Solve the problem.224)Newton’s Law of Cooling states that if a body with temperatureT1is placed in surroundings withtemperatureT0different fromT1, then the body will either cool or warm to temperature T(t) aftertime t, in minutes, whereT(t)=T0+(T1-T0)e-kt.A dish of lasagna baked at350°F is taken out of the oven into a kitchen that is71°F. After6minutes,the temperature of the lasagna is299.9°F. What will its temperature be14minutes after it was takenout of the oven? Round your answer to the nearest degree.224)A)242°FB)247°FC)234°FD)254°FAnswer:BExplanation:A)B)C)D)Evaluate to four decimal places using a calculator.225)e0.05225)A)0.1359B)1.0513C)2.9069D)1.6487Answer:BExplanation:A)B)C)D)Graph the function. Describe its position relative to the graph of the indicated basic function.96
226)f(x)=e-x+9; relative to f(x)=ex226)A)Moved right9units;reflected across the y-axisB)Moved left9units;reflected across the y-axisC)Reflected across the y-axis;moved right9unitsD)Reflected across the y-axis;moved left9unitsAnswer:CExplanation:A)B)C)D)97
Find the logarithm using the change-of-base formula.227)log60.88227)A)-0.0555B)-0.0713C)6.8182D)-14.0164Answer:BExplanation:A)B)C)D)Solve the logarithmic equation.228)log5(6x-1)=2228)A)log52+16B)20C)133D)263Answer:CExplanation:A)B)C)D)Sketch the graph of the function. Describe how the graph can be obtained from the graph of a basic logarithmic function.229)f(x)=ln x+4229)A)Shift y=ln x right4unitsB)Shift y=ln x up4units98
C)Shift y=ln x left4unitsD)Shift y=ln x down4unitsAnswer:BExplanation:A)B)C)D)Find a formula for the inverse of the function described below.230)32°Fahrenheit=0°Celsius. A function that converts temperatures inFahrenheitto those inCelsiusisf(x)=59(x-32).230)A)f-1(x)=95x+32B)f-1(x)=x+32C)f-1(x)=95x-32D)f-1(x)=59(x-32)Answer:AExplanation:A)B)C)D)Evaluate to four decimal places using a calculator.231)1e22231)A)0.1353B)0.0183C)54.5981D)7.3891Answer:BExplanation:A)B)C)D)99
Solve the problem.232)An initial investment of $14,000 is appreciated for11years in an account that earns8% interest,compounded semiannually. Find the amount of money in the account at the end of the period.232)A)$31,902.75B)$32,642.95C)$19,178.86D)$33,178.86Answer:DExplanation:A)B)C)D)233)Suppose that $1800is invested at5% interest, compounded quarterly. Find the function for theamount of money after t years.233)A)A(t)=1800(1.05)4tB)A(t)=1800(1.0125)tC)A(t)=1800(1.0125)4tD)A(t)=1800(1.05)tAnswer:CExplanation:A)B)C)D)Graph the function and its inverse using the same set of axes. Use any method.234)f(x)=log5x;f-1(x)=5×234)A)B)100
C)D)Answer:CExplanation:A)B)C)D)Evaluate to four decimal places using a calculator.235)e-0.90235)A)2.4596B)0.4493C)-2.4465D)0.4066Answer:DExplanation:A)B)C)D)Solve the exponential equation.236)64=45x·16×2236)A)-3,-4B)3,-12C)3,4D)-3,12Answer:DExplanation:A)B)C)D)Solve the problem.237)An initial investment of $1000 is appreciated for2years in an account that earns9% interest,compounded annually. Find the amount of money in the account at the end of the period.237)A)$1295.03B)$188.1C)$1090.00D)$1188.10Answer:DExplanation:A)B)C)D)101
Solve.238)Given thatloga5=1.609, andloga11=2.398, findloga155.238)A)-0.259B)-4.007C)0.259D)0.788Answer:BExplanation:A)B)C)D)Graph the function. Describe its position relative to the graph of the indicated basic function.239)f(x)=3-e-0.54x; relative to f(x)=ex239)A)Stretched horizontally;reflected across x-axis,reflected across y-axis;moved up3unit(s)B)Stretched horizontally;reflected across x-axis,reflected across y-axisMoved up3unit(s);102
C)Moved up3unit(s);stretched horizontally;reflected across the x-axisD)Stretched horizontally;reflected across the y-axis;moved up3unit(s)Answer:AExplanation:A)B)C)D)Find the domain and the vertical asymptote of the function.240)f(x)=ln x-4240)A)Domain: [0,Q); vertical asymptote: x=0B)Domain: (0,Q); vertical asymptote: x=0C)Domain: (-Q,Q); vertical asymptote: noneD)Domain: (-4,Q); vertical asymptote: x=-4Answer:BExplanation:A)B)C)D)Find the inverse of the function.241)f(x)=-3×241)A)f-1(x)=3xB)f-1(x)=3+xC)f-1(x)=-3xD)f-1(x)=-x3Answer:DExplanation:A)B)C)D)Solve the logarithmic equation.242)log x=-4242)A)0.0001B)-40C)-4D)-0.4Answer:AExplanation:A)B)C)D)103
Graph the function.243)y=18ex243)A)B)C)D)Answer:DExplanation:A)B)C)D)104
Express as a single logarithm and, if possible, simplify.244)23[ln (y2-4)-ln(y+2)]+ln (y+y)244)A)ln(y-2)2/3(y+y)B)ln(y-2)2/3(y-y)C)ln (y-2)(y-y)D)ln(y+2)2/3(y+y)Answer:AExplanation:A)B)C)D)Find the following using a calculator. Round to four decimal places.245)ln235245)A)5.4596B)86.7159C)0.1826D)2.3711Answer:AExplanation:A)B)C)D)105
Answer KeyTestname: C51)B2)A3)D4)C5)C6)A7)C8)C9)D10)D11)B12)B13)Answers may vary. One possible solution is:1.(f-1Hf)(x)=f-1(f(x))=f-1x-2/6=6x-2/6+2=x-2+2=x;2. (fHf-1)(x)=f(f-1(x))=f6x+2=6x+2-2/6=6x/6=x14)Graph f(x)=log x and g(x)=1 on the same set of axes. The solution is the first coordinate of the point of intersection ofthe two graphs.15)Answers may vary. Possible answer:Find two numbers a and b for which aJb and f(a)=f(b). Two such numbers are-2 and 2 because f(-2)=f(2)=-14.Thus, f is not one-to-one.16)Answers may vary. One possible solution is:1.(f-1Hf)(x)=f-1(f(x))=f-1×3-7=3×3-7+7=3×3=x;2. (fHf-1)(x)=f(f-1(x))=f3x+7=3x+73-7=x+7-7=x17)Assume that f(a)=f(b) for any numbers a and b in the domain of f.Since f(a)=6-12a and f(b)=6-12b, we have6-12a=6-12b-12a=-12ba=bThus if f(a)=f(b) then a=b and f is one-to-one.18)Since 3<Δ<4,23<2Δ<24; that is2Δis greater than23(or 8), but less than24(or 16). This is true because2xis anincreasing function.19)It excludes x=1ywhich is not a function.20)The quotientrspvcan be written as the product rs(p-1)(v-1)21)Answers may vary. Possible answer:Find two numbers a and b for which aJb and f(a)=f(b). Two such numbers are-3 and 3 because f(-3)=f(3)=1.Thus, f is not one-to-one.22)A difference of logarithms is not equal to the logarithm of the difference.log68-log6N=log68N23)Letloga5=x. Thenax=5. Replacing x withloga5, we havealoga5=5.106
Answer KeyTestname: C524)Answers may vary. Possible answer:Find two numbers a and b for which aJb and f(a)=f(b). Two such numbers are-1 and 1 because f(-1)=f(1)=3.Thus, f is not one-to-one.25)1.(f-1Hf)(x)=f-1(f(x))=f-1(3x+5)=(3x+5)3-5=x;2. (fHf-1)(x)=f(f-1(x))=f(x3-5)=3(x3-5)+5=x26)Consider f(x)=logax, or y=logax, where a is a positive constant other than 1. Thenay=x, and since a is positive, xmust be positive.27)2xis an exponential function because the variable is in the exponent. In f(x)=x2, the variable is not in the exponent;this is a polynomial function.28)83=83/2and 3/2<3, so83is larger.29)The most interest will be earned in the fifth year, because the amount on which the interest is paid is greatest at the endof the fourth year.30)124731)1.(f-1Hf)(x)=f-1(f(x))=f-172x=2772x=x;2. (fHf-1)(x)=f(f-1(x))=f27x=7227x=x32)Since the bases are the same, the base with the larger exponent is the larger number. Thus,Δ2.4is larger.33)Reflect the graph of f(x)=ln x across the line y=x and then translate it to the right one unit.34)Answers may vary. Possible answer:Find two numbers a and b for which aJb and f(a)=f(b). Two such numbers are-1 and 1 because f(-1)=f(1)=2.Thus, f is not one-to-one.35)1.(f-1Hf)(x)=f-1(f(x))=f-1x+49=9x+49-4=x;2. (fHf-1)(x)=f(f-1(x))=f(9x-4)=9x-4+49=x36)Assume that f(a)=f(b) for any numbers a and b in the domain of f.Since f(a)=a3-4 and f(b)=b3-4, we havea3-4=b3-4a3=b3a=bThus if f(a)=f(b) then a=b and f is one-to-one.37)A sum of logarithms is not equal to a logarithm of a sum.log32+log3M=log32M38)Reflect the graph of f(x)=exacross the line y=x and then translate it up one unit.39)Answers may vary. One possible solution is:1.(f-1Hf)(x)=f-1(f(x))=f-11+x/x=11+xx-1=11+x-xx=11x=x;2. (fHf-1)(x)=f(f-1(x))=f1x-1=1+1x-11x-1=1x-1+1x-11x-1=1×1=x107
Answer KeyTestname: C540)log28=3 andlog216=4. Since 8<13<16, then 3<log213<4. This is true becauselog2x is an increasingfunction.41)Assume that f(a)=f(b) for any numbers a and b in the domain of f.Since f(a)=3a+5 and f(b)=3b+5, we have3a+5=3b+53a=3ba=bThus if f(a)=f(b) then a=b and f is one-to-one.42)Answers may vary. One possible solution is:1.(f-1Hf)(x)=f-1(f(x))=f-1-67x=-76-67x=x;2. (fHf-1)(x)=f(f-1(x))=f-76x=-67-76x=x43)A logarithm of a product is not equal to a product of logarithms.log43y=log43+log4y44)Assume that f(a)=f(b) for any numbers a and b in the domain of f.Since f(a)=3a-1 and f(b)=3b-1, we have3a-1=3b-13a=3ba=bThus if f(a)=f(b) then a=b and f is one-to-one.45)Answers may vary. One possible solution is:1.(f-1Hf)(x)=f-1(f(x))=f-1x+7/5=5x+7/5-7=x+7-7=x;2. (fHf-1)(x)=f(f-1(x))=f5x-7=5x-7+7/5=5x/5=x46)A47)A48)B49)C50)B51)D52)A53)C54)C55)D56)C57)B58)D59)B60)C61)D62)C63)D64)D65)B66)D67)D68)A108
Answer KeyTestname: C569)A70)C71)B72)A73)D74)B75)A76)B77)B78)A79)B80)A81)C82)B83)C84)A85)A86)B87)D88)B89)D90)A91)B92)D93)B94)D95)D96)D97)C98)D99)D100)C101)D102)D103)C104)A105)A106)B107)D108)B109)B110)B111)C112)B113)C114)A115)B116)B117)B118)A109
Answer KeyTestname: C5119)D120)C121)C122)D123)B124)A125)C126)D127)C128)D129)D130)A131)B132)A133)C134)A135)B136)A137)C138)B139)D140)B141)B142)C143)C144)B145)D146)A147)D148)C149)A150)D151)D152)A153)D154)D155)A156)D157)C158)A159)B160)A161)C162)A163)C164)A165)B166)A167)A168)D110
Answer KeyTestname: C5169)D170)D171)B172)A173)D174)C175)A176)B177)A178)C179)B180)A181)C182)A183)B184)C185)C186)B187)D188)D189)D190)A191)A192)B193)A194)D195)B196)C197)D198)D199)A200)D201)C202)A203)B204)A205)B206)C207)A208)C209)D210)B211)D212)D213)B214)D215)D216)A217)B218)B111
Answer KeyTestname: C5219)B220)D221)C222)C223)C224)B225)B226)C227)B228)C229)B230)A231)B232)D233)C234)C235)D236)D237)D238)B239)A240)B241)D242)A243)D244)A245)A112

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