Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Calculus: Early Transcendentals
If f(x)=x+2x and g(u)=u+2u, is it true that f = g?If f(x)=x2xx1andg(x)=x is it true that f = g?The graph of a function f is given. (a) State the value of f(1). (b) Estimate the value of f(1). (c) For what values of x is f(x) = 1? (d) Estimate the value of x such that f(x) = 0. (e) State the domain and range of f. (f) On what interval is f increasing?The graphs of f and g are given. (a) State the values of f(4) and g(3). (b) For what values of x is f(x) = g(x)? (c) Estimate the solution of the equation .f(x) = 1. (d) On what interval is f decreasing? (c) State the domain and range of f. (f) State the domain and range of g.Figure 1 was recorded by an instrument operated by the California Department of Mines and Geology at the University Hospital of the University of Southern California in Los Angeles. Use it to estimate the range of the vertical ground acceleration function at USC during the Northridge earthquake.Determine whether the curve is the graph of a function of x. If it is, state the domain and range of the function. 7.Determine whether the curve is the graph of a function of x. If it is, state the domain and range of the function. 8.Determine whether the curve is the graph of a function of x. If it is, state the domain and range of the function. 9.Determine whether the curve is the graph of a function of x. If it is, state the domain and range of the function. 10.Shown is a graph of the global average temperature T during the 20th century. Estimate the following. (a) The global average temperature in 1950 (b) The year when the average temperature was 14.2C (c) The year when the temperature was smallest? Largest? (d) The range of T Source: Adapted from Globe and Mail [Toronto]. 5 Dec. 2009. Print.Trees grow faster and form wider rings in warm years and grow more slowly and form narrower rings in cooler years. The figure shows ring widths of a Siberian pine from 1500 to 2000. (a) What is the range of the ring width function? (b) What does the graph tend to say about the temperature of the earth? Does the graph reflect the volcanic eruptions of the mid-19th century?You put some ice cubes in a glass, fill the glass with cold water, and then let the glass sit on a table. Describe how the temperature of the water changes as time passes. Then sketch a rough graph of the temperature of the water as a function of the elapsed time.Three runners compete in a 100-meter race. The graph depicts the distance run as a function of time for each runner. Describe in words what the graph tells you about this race. Who won the race? Did each runner finish the race?The graph shows the power consumption for a day in September in San Francisco. (P is measured in megawatts; t is measured in hours starting at midnight.) (a) What was the power consumption at 6 AM? At 6 PM? (b) When was the power consumption the lowest? When was it the highest? Do these times seem reasonable?Sketch a rough graph of the number of hours of daylight as a function of the time of year.Sketch a rough graph of the outdoor temperature as a function of time during a typical spring day.Sketch a rough graph of the market value of a new car as a function of time for a period of 20 years. Assume the car is well maintained.Sketch the graph of the amount of a particular brand of coffee sold by a store as a function of the price of the coffee.You place a frozen pie in an oven and bake it for an hour. Then you take it out and let it cool before eating it. Describe how the temperature of the pie changes as time passes. Then sketch a rough graph of the temperature of the pie as a function of time.A homeowner mows the lawn every Wednesday afternoon. Sketch a rough graph of the height of the grass as a function of time over the course of a four-week period.An airplane takes off from an airport and lands an hour later at another airport, 400 miles away. If t represents the time in minutes since the plane has left the terminal building, let x(t) be the horizontal distance traveled and y(t) be the altitude of the plane. (a) Sketch a possible graph of x(t). (b) Sketch a possible graph of y(t). (c) Sketch a possible graph of the ground speed. (d) Sketch a possible graph of the vertical velocity.Temperature readings T (in F) were recorded every two hours from midnight to 2:00 PM in Atlanta on June 4, 2013. The time t was measured in hours from midnight. t 0 2 4 6 8 10 12 14 T 74 69 68 66 70 78 82 86 (a) Use the readings to sketch a rough graph of T as a function of t. (b) Use your graph to estimate the temperature at 9:00 AM.Researchers measured the blood alcohol concentration (BAC) of eight adult male subjects after rapid consumption of 30 mL of ethanol (corresponding to two standard alcoholic drinks). The table shows the data they obtained by averaging the BAC (in mg/mL) of the eight men. (a) Use the readings to sketch the graph of the BAC as a function of t. (b) Use your graph to describe how the effect of alcohol varies with time. t (hours) BAC t (hours) BAC 0 0 1.75 0.22 0.2 0.25 2.0 0.18 0.5 0.41 2.25 0.15 0.75 0.40 2.5 0.12 1.0 0.33 3.0 0.07 1.25 0.29 3.5 0.03 1.5 0.24 4.0 0.01 Source: Adapted from P. Wilkinson et al., Pharmacokinetics of Ethanol after Oral Administration in the Fasting Stale, Journal of Pharmacokinetics and Biopharmaceutics 5 (1977): 20724.If f(x) = 3x2 x + 2, find f(2), f(2), f(a), f(a), f(a + 1), 2f(a), f(2a), f(a2), [f(a)]2, and f(a + h).A spherical balloon with radius r inches has volume V(r)=43r3. Find a function that represents the amount of air required to inflate the balloon from a radius of r inches to a radius of r + 1 inches.Evaluate the difference quotient for the given function. Simplify your answer. 27. f(x)=4+3xx2,f(3+h)f(3)hEvaluate the difference quotient for the given function. Simplify your answer. 28. f(x)=x3,f(a+h)f(a)hEvaluate the difference quotient for the given function. Simplify your answer. 29. f(x)=1x, f(x)f(a)xaEvaluate the difference quotient for the given function. Simplify your answer. 30. f(x)=x+3x+1, f(x)f(1)x1Find the domain of the function. 31. f(x)=x+4x29Find the domain of the function. 32. f(x)=2x35x2+x6Find the domain of the function. 33. f(t)=2t13Find the domain of the function. 34. g(t)=3t2+tFind the domain of the function. 35. h(x)=1x25x4Find the domain of the function. 36. f(u)=u+11+1u+1Find the domain of the function. 37. F(p)=2pFind the domain and range and sketch the graph of the function h(x)=4x2.Find the domain and sketch the graph of the function. 39. f(x)=1.6x2.4Find the domain and sketch the graph of the function. 40. g(t)=t21t+1Evaluate f(3), f(0), and f(2) for the piecewise defined function. Then sketch the graph of the function. 41. f(x)={x+2ifx01xifx0Evaluate f(3), f(0), and f(2) for the piecewise defined function. Then sketch the graph of the function. 42. f(x)={312xifx22x5ifx2Evaluate f(3), f(0), and f(2) for the piecewise defined function. Then sketch the graph of the function. 43. f(x)={x+1ifx1x2ifx1Evaluate f(3), f(0), and f(2) for the piecewise defined function. Then sketch the graph of the function. 44. f(x)={1ifx172xifx1Sketch the graph of the function. 45. f(x) = x + |x|Sketch the graph of the function. 46. f(x) = |x + 2|Sketch the graph of the function. 47. g(t) = |1 3t|Sketch the graph of the function. 48. h(t) = |t| + |t + 1|Sketch the graph of the function. 49. f(x)={|x|if|x|11if|x|1Sketch the graph of the function. 50. g(x) = ||x| 1|Find an expression for the function whose graph is the given curve. 51. The line segment joining the points (1, 3) and (5, 7)Find an expression for the function whose graph is the given curve. 52. The line segment joining the points (5, 10) and (7, 10)Find an expression for the function whose graph is the given curve. 53. The bottom half of the parabola x + (y 1)2 = 0Find an expression for the function whose graph is the given curve. 54. The top half of the circle x2 + (y 2)2 = 4Find an expression for the function whose graph is the given curve. 55.Find an expression for the function whose graph is the given curve. 56.Find a formula for the described function and state its domain. 57. A rectangle has perimeter 20 m. Express the area of the rectangle as a function of the length of one of its sides.Find a formula for the described function and state its domain. 58. A rectangle has area 16 m2. Express the perimeter of the rectangle as a function of the length of one of its sides.Find a formula for the described function and state its domain. 59. Express the area of an equilateral triangle as a function of the length of a side.Find a formula for the described function and state its domain. 60. A closed rectangular box with volume 8 ft3 has length twice the width. Express the height of the box as a function of the width.Find a formula for the described function and state its domain. 61. An open rectangular box with volume 2 m3 has a square base. Express the surface area of the box as a function of the length of a side of the base.A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 30 ft, express the area A of the window as a function of the width x of the window.A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in. by 20 in. by cutting out equal squares of side x at each comer and then folding up the sides as in the figure. Express the volume V of the box as a function of x.A cell phone plan has a basic charge of 35 a month. The plan includes 400 free minutes and charges 10 cents for each additional minute of usage. Write the monthly cost C as a function of the number x of minutes used and graph C as a function of x for 0 x 600.In a certain state the maximum speed permitted on freeways is 65 mi/h and the minimum speed is 40 mi/h. The fine for violating these limits is 15 for every mile per hour above the maximum speed or below the minimum speed. Express the amount of the fine F as a function of the driving speed x and graph F(x) for 0 x 100.An electricity company charges its customers a base rate of 10 a month, plus 6 cents per kilowatt-hour (kWh) for the first 1200 kWh and 7 cents per kWh for all usage over 1200 kWh. Express the monthly cost E as a function of the amount x of electricity used. Then graph the function E for 0 x 2000.In a certain country, income tax is assessed as follows. There is no tax on income up to10,000. Any income over 10,000 is taxed at a rate of 10%, up to an income of 20,000. Any income over 20,000 is taxed at 15%. (a) Sketch the graph of the tax rate R as a function of the income I. (b) How much tax is assessed on an income of 14,000? On 26,000? (c) Sketch the graph of the total assessed tax T as a function of the income I.The functions in Example 10 and Exercise 67 are called step function because their graphs look like stairs. Give two other examples of step functions thru arise in everyday life.Graphs of f and g are shown. Decide whether each function is even, odd, or neither. Explain your reasoning. 69.Graphs of f and g are shown. Decide whether each function is even, odd, or neither. Explain your reasoning. 70.(a) If the point (5, 3) is on the graph of an even function, what other point must also be on the graph? (b) If the point (5, 3) is on the graph of an odd function, what other point must also be on the graph?A function f has domain [5, 5] and a portion of its graph is shown. (a) Complete the graph of f if it is known that f is even. (b) Complete the graph of f if it is known that f is odd.Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. 73. f(x)=xx2+1Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. 74. f(x)=x2x4+1Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. 75. f(x)=xx+1Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. 76. f(x) = x |x|Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. 77. f(x) = 1 + 3x2 x4Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. 78. f(x)=1+3x3x5If f and g are both even functions, is f + g even? If f and g are both odd functions, is f + g odd? What if f is even and g is odd? Justify your answers.If f and g are both even functions, is the product of fg even? If f and g are both odd functions, is fg odd? What if f is even and g is odd? Justify your answers.Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function. 1. (a) f(x)=log2x (b) g(x)=x4 (c) h(x)=2x31x2 (d) u(t)=11.1t+2.54t2 (e) v(t)=5t (f) w()=sincos2Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function. 2. (a) y=x (b) y=x (c) y=xx(2x3) (d) y=tantcost (e) y=s1+s (f) y=x311+x3Match each equation with its graph. Explain your choices. (Don't use a computer or graphing calculator.) 3. (a) y=x2 (b) y=x5 (c) y=x8Match each equation with its graph. Explain your choices. (Don't use a computer or graphing calculator.) 4. (a) y=3x (b) y=3x (c) y=x3 (d) y=x3Find the domain of the function. 5. f(x)=cosx1sinxFind the domain of the function. 6. g(x)=11tanx(a) Find an equation for the family of linear functions with slope 2 and sketch several members of the family. (b) Find an equation for the family of linear functions such that f(2) = 1 and sketch several members of the family. (c) Which function belongs to both families?What do all members of the family of linear functions f(x) = 1 + m(x + 3) have in common? Sketch several members of the family.What do all members of the family of linear functions f(x)=cxhave in common? Sketch several members of the family.Find expressions for the quadratic functions whose graphs are shown.Find an expression for a cubic function f if f(1)=6 and f(1)=f(0)=f(2)=0.Recent studies indicate that the average surface temperature of the earth has been rising steadily. Some scientists have modeled the temperature by the linear function T=0.02t+8.50, where T is temperature in C and t represents years since 1900. (a) What do the slope and T-intercept represent? (b) Use the equation to predict the average global surface temperature in 2100.If the recommended adult dosage for a drug is D (in mg), then to determine the appropriate dosage c for a child of age a, pharmacists use the equation c = 0.0417D(a + 1). Suppose the dosage for an adult is 200 mg. (a) Find the slope of the graph of c. What does it represent? (b) What is the dosage for a newborn?The manager of a weekend flea market knows from past experience that if he charges x dollars for a rental space at the market, then the number y of spaces he can rent is given by the equationy=2004x. (a) Sketch a graph of this linear function. (Remember that the rental charge per space and the number of spaces rented can't be negative quantities.) (b) What do the slope, the y-intercept, and the x-intercept of the graph represent?The relationship between the Fahrenheit (F) and Celsius (C) temperature scales is given by the linear function F=95C+32. (a) Sketch a graph of this function. (b) What is the slope of the graph and what does it represent? What is the F-intercept and what does it represent?Jason leaves Detroit at 2:00 PM and drives at a constant speed west along I-94. He passes Ann Arbor, 40 mi from Detroit, at 2:50 PM. (a) Express the distance traveled in terms of the time elapsed. (b) Draw the graph of the equation in part (a). (c) What is the slope of this line? What does it represent?Biologists have noticed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 113 chirps per minute at 70F and 173 chirps per minute at 80F. (a) Find a linear equation that models the temperature T as a function of the number of chirps per minute N. (b) What is the slope of the graph? What does it represent? (c) If the crickets are chirping at 150 chirps per minute, estimate the temperature.The manager of a furniture factory finds that it costs 2200 to manufacture 100 chairs in one day and 4800 to produce 300 chairs in one day. (a) Express the cost as a function of the number of chairs produced, assuming that it is linear. Then sketch the graph. (b) What is the slope of the graph and what does it represent? (c) What is the y-intercept of the graph and what does it represent?At the surface of the ocean, the water pressure is the same as the air pressure above the water, 15 lb/in2. Below the surface, the water pressure increases by 4.34 lb/in2 for every 10 ft of descent. (a) Express the water pressure as a function of the depth below the ocean surface. (b) At what depth is the pressure 100 lb/in2?The monthly cost of driving a car depends on the number of miles driven. Lynn found that in May it cost her 380 to drive 480 mi and in June it cost her 460 to drive 800 mi. (a) Express the monthly cost C as a function of the distance driven d. assuming that a linear relationship gives a suitable model. (b) Use part (a) to predict the cost of driving 1500 miles per month. (c) Draw the graph of the linear function. What does the slope represent? (d) What does the C-intercept represent? (e) Why does a linear function give a suitable model in this situation?For each scatter plot, decide what type of function you might choose as a model for the data. Explain your choices. 21.For each scatter plot, decide what type of function you might choose as a model for the data. Explain your choices. 22.Suppose the graph of f is given. Write equations for the graphs that are obtained from the graph of f as follows. (a) Shift 3 units upward. (b) Shift 3 units downward. (c) Shift 3 units to the right. (d) Shift 3 units to the left. (e) Reflect about the x-axis. (f) Reflect about the y-axis. (g) Stretch vertically by a factor of 3. (h) Shrink vertically by a factor of 3.Explain how each graph is obtained from the graph ofy=f(x). (a) y=f(x)+8 (b) y=f(x+8) (c) y=8f(x) (d) y=f(8x) (e) y=f(x)1 (f) y=8f(18x)The graph of y=f(x) is given. Match each equation with its graph and give reasons for your choices. (a) y=f(x4) (b) y=f(x)+3 (c) y=13f(x) (d) y=f(x+4) (e) y=2f(x+6)The graph of f is given. Draw the graphs of the following functions. (a) y=f(x)3 (b) y=f(x+1) (c) y=12f(x) (d) y=f(x)The graph of f is given. Use it to graph the following functions. (a) y=f(2x) (b) y=f(12x) (c) y=f(x) (d) y=f(x)The graph of y=3xx2 is given. Use transformations to create a function whose graph is as shown.The graph of y=3xx2 is given. Use transformations to create a function whose graph is as shown.(a) How is the graph of y = 2 sin x related to the graph of y = sin x? Use your answer and Figure 6 to sketch the graph of y = 2 sin x. (b) How is the graph of y=1+x related to the graph of y=x? Use your answer and Figure 4(a) to sketch the graph of y=1+x.Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. 9. y=x210EGraph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. 11.y=x3+1Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. 12. y=11xGraph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. 13. y=2cos3xGraph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. 14. y=2x+1Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. 15.y=x24x+5Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. 16. y=1+sinxGraph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. 17. y=2x18EGraph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. 19. y=sin(12x)Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. 20. y=|x|221E22E23E24EThe city of New Orleans is located at latitude 30N. Use Figure 9 to find a function that models the number of hours of daylight at New Orleans as a function of the time of year. To check the accuracy of your model, use the fact that on March 31 the sun rises at 5:51 AM and sets at 6:18 PM in New Orleans.A variable star is one whose brightness alternately increases and decreases. For the most visible variable star, Delta Cephei, the time between periods of maximum brightness is 5.4 days, the average brightness (or magnitude) of the star is 4.0, and its brightness varies by 0.35 magnitude. Find a function that models the brightness of Delta Cephei as a function of time.Some of the highest tides in the world occur in the Bay of Fundy on the Atlantic Coast of Canada. At Hopewell Cape the water depth at low tide is about2.0 m and at high tide it is about 12.0 m. The natural period of oscillation is about 12 hours and on June 30, 2009. high tide occurred at 6:45 AM. Find a function involving the cosine, function that models the water depth D(t) (in meters) as a function of time t (in hours after midnight) on that day.In a normal respiratory cycle the volume of air that moves into and out of the lungs is about 500 mL. The reserve and residue volumes of air that remain in the lungs occupy about 2000 mL and a single respiratory cycle for an average human takes about 4 seconds. Find a model for the total volume of air V(t) in the lungs as a function of time.(a) How is the graph of y=f(|x|)related to the graph of f? (b) Sketch the graph of y=sin|x|. (c) Sketch the graph of y=|x|.Use the given graph of f to sketch the graph of y = 1/f(x). Which features of f are the most important in sketching y = 1/f(x)? Explain how they are used.31EFind (a) f + g, (b) f g, (c) fg, and (d) f/g and state their domains. 32. f(x)=3x,g(x)=x21Find the functions (a) fg, (b) gf, (c) ff, and (d) gg and their domains. 33. f(x)=3x+5,g(x)=x2+xFind the functions (a) fg, (b) gf, (c) ff, and (d) gg and their domains. 34. f(x)=x32, g(x)=14xFind the functions (a) fg, (b) gf, (c) ff, and (d) gg and their domains. 35. f(x)=x+1,g(x)=4x336EFind the functions (a) f g, (b) g f, (c) f f, and (d) g g and their domains. 37. f(x)=x+1x,g(x)=x+1x+2Find the functions (a) f g, (b) g f, (c) f f, and (d) g g and their domains. 38. f(x)=x1+x, g(x) = sin 2xFind f g h. 39. f(x) = 3x 2, g (x) = sin x, h(x) = x240EFind f g h. 41.f(x)=x3, g(x) = x2, h(x) = x3 + 2Find f g h. 42. f(x) = tan x, g(x)=xx1,h(x)=x3Express the function in the form f g. 43. F(x) = (2x + x2)4Express the function in the form f g. 44. F(x) = cos2x45E46EExpress the function in the form f g. 47. v(t) = sec(t2) tan(t2)48E49E50E51EUse the table to evaluate each expression. (a) f(g(1)) (b) g(f(1)) (c) f(f(1)) (d) g(g(1)) (e) (g f)(3) (f) (f g)(6)Use the given graphs of f and g to evaluate each expression, or explain why it is undefined. (a) f(g(2)) (b) g(f(0)) (c) (f g)(0) (d) (g f)(6) (e) (g g)(2) (f) (f f)(4)Use the given graphs of f and g to estimate the value of f(g(x)) for x = 5, 4, 3,. . . , 5. Use these estimates to sketch a rough graph of f g.A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm/s. (a) Express the radius r of this circle as a function of the time t (in seconds). (b) If A is the area of this circle as a function of the radius, find A r and interpret it.A spherical balloon is being inflated and the radius of the balloon is increasing at a rate of 2 cm/s. (a) Express the radius r of the balloon as a function of the time t (in seconds). (b) If V is the volume of the balloon as a function of the radius, find V r and interpret it.57EAn airplane is flying at a speed of 350 mi/h at an altitude of one mile and passes directly over a radar station at time t = 0. (a) Express the horizontal distance d (in miles) that the plane has flown as a function of t. (b) Express the distance s between the plane and the radar station as a function of d. (c) Use composition to express s as a function of t.The Heaviside function H is defined by It is used in the study of electric circuits to represent the sudden surge of electric current, or voltage, when a switch is instantaneously turned on. (a) Sketch the graph of the Heaviside function. (b) Sketch the graph of the voltage V(t) in a circuit if the switch is turned on at time t = 0 and 120 volts are applied instantaneously to the circuit. Write a formula for V(t) in terms of H(t). (c) Sketch the graph or the voltage V(t) in a circuit if the switch is turned on at time t = 5 seconds and 240 volts are applied instantaneously to the circuit. Write a formula for V(t) in terms of H(t). (Note that starting at t = 5 corresponds to a translation.)The Heaviside function defined in Exercise 59 can also be used to define the ramp function y = ctH(t), which represents a gradual increase in voltage or current in a circuit. (a) Sketch the graph of the ramp function y = tH(t). (b) Sketch the graph of the voltage V(t) in a circuit if the switch is turned on at time t = 0 and the voltage is gradually increased to 120 volts over a 60-second time interval. Write a formula for V(t) in terms of H(t) for t 60. (c) Sketch the graph of the voltage V(t) in a circuit if the switch is turned on at time t = 7 seconds and the voltage is gradually increased to 100 volts over a period of 25 seconds. Write a formula for V(t) in terms of H(t) for t 32.Let f and g be linear functions with equations f(x) = m1x + b1, and g(x) = m2x + b2. Is f g also a linear function? If so, what is the slope of its graph?62E(a) If g(x) = 2x + 1 and h(x) = 4x2 + 4x + 7, find a function f such that f g = h. (Think about what operations you would have to perform on the formula for g to end up with the formula for h.) (b) If f(x) = 3x + 5 and h(x) = 3x2 + 3x + 2, find a function g such that f g = h.If f(x) = x + 4 and h(x) = 4x 1, find a function g such that g f = h.Suppose g is an even function and let h =f g. Is h always an even function?Suppose g is an odd function and let h = f g. Is h always an odd function? What it f is odd? What it f is even?Use the Law of Exponents to rewrite and simplify the expression. 1. (a) 4328 (b) 1x43Use the Law of Exponents to rewrite and simplify the expression. 2. (a) 84/3 (b) x(3x2)3Use the Law of Exponents to rewrite and simplify the expression. 3. (a) b8(2b)4 (b) (6y3)42y5Use the Law of Exponents to rewrite and simplify the expression. 4. (a) x2nx3n1xn+2 (b) abab3(a) Write an equation that defines the exponential function with base b 0. (b) What is the domain of this function? (c) If b 1, what is the range of this function? (d) Sketch the general shape or the graph of the exponential function for each of the following cases. (i) b 1 (ii) b = 1 (iii) 0 b 1(a) How is the number e defined? (b) What is an approximate value for e? (c) What is the natural exponential function?Graph the given functions on a common screen. How are these graphs related? 7. y = 2x, y = ex, y = 5x, y = 20xGraph the given functions on a common screen. How are these graphs related? 8. y=ex,y=ex,y=8x,y=8xGraph the given functions on a common screen. How are these graphs related? 9. y = 3x, y = 10x, y=(13)x, y=(110)xGraph the given functions on a common screen. How are these graphs related? 10. y = 0.9x, y = 0.6x, y = 0.3x, y = 0.1xMake a rough sketch of the graph of the function. Do not use a calculator. Just use the graphs given in Figures 3 and 13 and, if necessary, the transformations of Section 1.3. 11. y = 4x 1Make a rough sketch of the graph of the function. Do not use a calculator. Just use the graphs given in Figures 3 and 13 and, if necessary, the transformations of Section 1.3. 12. y = (0.5)x1Make a rough sketch of the graph of the function. Do not use a calculator. Just use the graphs given in Figures 3 and 13 and, if necessary, the transformations of Section 1.3. 13. y = 2xMake a rough sketch of the graph of the function. Do not use a calculator. Just use the graphs given in Figures 3 and 13 and, if necessary, the transformations of Section 1.3. 14. y = e |x|Make a rough sketch of the graph of the function. Do not use a calculator. Just use the graphs given in Figures 3 and 13 and, if necessary, the transformations of Section 1.3. 15. y=112exMake a rough sketch of the graph of the function. Do not use a calculator. Just use the graphs given in Figures 3 and 13 and, if necessary, the transformations of Section 1.3. 16. y = 2(1 ex)Starting with the graph of y = ex, write the equation of the graph that results from (a) shifting 2 units downward. (b) shifting 2 units 10 the right. (c) reflecting about the x-axis. (d) reflecting about the y-axis. (c) reflecting about the x-axis and then about the y-axis.Starting with the graph of y = ex, find the equation of the graph that results from (a) reflecting about the line y = 4. (b) reflecting about the line x = 2.Find the domain of each function. 19. (a) f(x)=1ex21e1x2 (b) f(x)=1+xecosxFind the domain of each function. 20. (a) g(t)=10r100 (b) g(t) = sin(et 1)Find the exponential function f(x) = Cb2 whose graph is given.Find the exponential function f(x) = Cbx whose graph is given.If f(x) = 5x, show that f(x+h)f(x)h=5x(5h1h)Suppose you are offered a job that lasts one month. Which of the following method of payment do you prefer? I. One million dollars at the end of the month. ll. One cent on the first day of the month, two cents on the second day, four cents on the third day, and, in general 2n-1 cents on the nth day.25ECompare the functions f(x) = x5and g(x) = 5x by graphing both functions in several viewing rectangles. Find all points of intersection of the graphs correct to one decimal place. Which function grows more rapidly when x is large?Compare the functions f(x) = x10 and g(x) = ex by graphing both f and g in several viewing rectangles. When does the graph of g finally surpass the graph of f?Use a graph to estimate the values of x such that ex 1,000,000,000.A bacteria culture starts with 500 bacteria and doubles in size every half hour. (a) How many bacteria are there after 3 hours? (b) How many bacteria are there after t hours? (c) How many bacteria are there after 40 minutes? (d) Graph the population function and estimate the time for the population to reach 100,000.The half-life of bismuth-210, 210Bi, is 5 days. (a) If a sample has a mass of 200 mg, find the amount remaining after 15 days. (b) Find the amount remaining after t days. (c) Estimate the amount remaining after 3 weeks. (d) Use a graph to estimate the time required for the mass to be reduced to 1 mg.An isotope of sodium, 24Na, has a half-life of 15 hours. A sample of this isotope has mass 2 g. (a) Find the amount remaining after 60 hours. (b) Find the amount remaining after t hours. (c) Estimate the amount remaining after 4 days. (d) Use a graph to estimate the time required for the mass to he reduced to 0.01 g.33EAfter alcohol is fully absorbed into the body, it is metabolized with a half-life of about 1.5 hours. Suppose you have had three alcoholic drinks and an hour later, at midnight, your blood alcohol concentration (BAC) is 0.6 mg/ mL. (a) Find an exponential decay model for your BAC t hours after midnight. (b) Graph your BAC and use the graph to determine when your BAC is 0.08 mg/mL. Source: Adapted from P. Wilkinson et al., Pharmacokinetics of Ethanol after Oral Administration in the Fasting State, Journal of Pharmacokinetics and Biopharmaceutics 5 (1977): 20724.Use a graphing calculator with exponential regression capability to model the population of the world with the data from 1950 to 2010 in Table 1 on page 49. Use the model to estimate the population in 1993 and to predict the population in the year 2020.The table gives the population of the United States, in millions, for the years 1900- 2010. Use a graphing calculator with exponential regression capability to model the US population since 1900. Use the model to estimate the population in 1925 and to predict the population in the year 2020.If you graph the function f(x)=1e1/x1+e1/x you' ll see that f appears to be an odd function. Prove it.Graph several members of the family of functions f(x)=11+aebx where a 0. How does the graph change when b changes? How does it change when a changes?(a) What is a one-to-one function? (b) How can you tell from the graph of a function whether it is one-to-one?(a) Suppose f is a one-to-one function with domain A and range B. How is the inverse function f-1 defined? What is the domain of f-1? What is the range of f-1? (b) If you are given a formula for f, how do you find a formula for f-1? (c) If you are given the graph of f, how do you find the graph of f-1?A function is given by a table of values, a graph, a formula or a verbal description. Determine whether it is one-to-one.A function is given by a table of values, a graph, a formula or a verbal description. Determine whether it is one-to-one.A function is given by a table of values, a graph, a formula or a verbal description. Determine whether it is one-to-one.A function is given by a table of values, a graph, a formula or a verbal description. Determine whether it is one-to-one.A function is given by a table of values, a graph, a formula or a verbal description. Determine whether it is one-to-one.A function is given by a table of values, a graph, a formula or a verbal description. Determine whether it is one-to-one. 8.A function is given by a table of values, a graph, a formula or a verbal description. Determine whether it is one-to-one. 9. f(x) = 2x - 3A function is given by a table of values, a graph, a formula or a verbal description. Determine whether it is one-to-one. 10. f(x) = x4 16A function is given by a table of values, a graph, a formula or a verbal description. Determine whether it is one-to-one. 11. g(x) = 1 sin xA function is given by a table of values, a graph, a formula or a verbal description. Determine whether it is one-to-one. 12. g(x)=x3A function is given by a table of values, a graph, a formula or a verbal description. Determine whether it is one-to-one. 13. f(t) is the height of a football t seconds after kickoff.A function is given by a table of values, a graph, a formula or a verbal description. Determine whether it is one-to-one. 14. f(t) is your height at age t.Assume that f is a one-to-one function. (a) If f(6) = 17, what is f1(17)? (b) If f1(3) = 2, what is f(2)?If f(x) = x5 + x3 + x, find f1(3) and f(f 1(2)).If g(x) = 3 + x + ex, find g1(4).The graph of f is given. (a) Why is f one-to-one? (b) What are the domain and range of f1? (c) What is the value of f1(2)? (d) Estimate the value of f1(0).The formula C=59(F32), where F 459.67, expresses the Celsius temperature C as a function of the Fahrenheit temperature F. Find a formula for the inverse function and interpret it. What is the domain of the inverse function?In the theory of relativity, the mass of a particle with speed is m=f()=m012/c2 where m0 is the rest mass of the particle and c is the speed of light in a vacuum. Find the inverse function of f and explain its meaning.Find a formula for the inverse of the function. 21. f(x)=1+2+3xFind a formula for the inverse of the function. 22. f(x)=4x12x+3Find a formula for the inverse of the function. 23. f(x) = e2x1Find a formula for the inverse of the function. 24. y=x2x,x12Find a formula for the inverse of the function. 25. y = ln(x + 3)Find a formula for the inverse of the function. 26. y=1ex1+exFind an explicit formula for f1 and use it to graph f1, f, and the line y = x on the same screen. To check your work, see whether the graphs of f and f1 are reflections about the line. 27. f(x)=4x+3Find an explicit formula for f1 and use it to graph f1, f, and the line y = x on the same screen. To check your work, see whether the graphs of f and f1 are reflections about the line. 28. f(x) = 1 + ex29EUse the given graph of f to sketch the graph of f1. 30.Let f(x)=1x2,0x1. (a) Find f1. How is it related to f? (b) Identify the graph of f and explain your answer to part (a).32E33E(a) What is the natural logarithm? (b) What is the common logarithm? (c) Sketch the graphs of the natural logarithm function and the natural exponential function with a common set of axes.Find the exact value of each expression. 35. (a) log2 32 (b) log8 2Find the exact value of each expression. 35. (a) log51125 (b) ln(1/e2)Find the exact value of each expression. 37. (a) log10 40 + log10 2.5 (b) log8 60 log8 3 log8 5Find the exact value of each expression. 38. (a) eln2 (b) eln(lne3)Express the given quantity as a single logarithm. 39. ln 10 + 2 ln 5Express the given quantity as a single logarithm. 40. ln b + 2 ln c 3 ln dExpress the given quantity as a single logarithm. 41. 13ln(x+2)3+12[lnxln(x2+3x+2)2]Use Formula 10 to evaluate each logarithm correct to six decimal places. (a) log5 10 (b) log3 57Use Formula 10 to graph the given functions on a common screen. How are these graphs related? 43. y=log1.5x,y=lnx,y=log10x,y=log50xUse Formula 10 to graph the given functions on a common screen. How are these graphs related? 44. y=lnx,y=log10x,y=ex,y=10xSuppose that the graph of y = log2 x is drawn on a coordinate grid where the unit of measurement is an inch. How many miles to the right of the origin do we have to move before the height of the curve reaches 3 ft?46E47E48E(a) What are the domain and range of f? (b) What is the x-intercept of the graph of f? (c) Sketch the graph of f. 49. f(x) = ln x + 2(a) What are the domain and range of f? (b) What is the x-intercept of the graph of f? (c) Sketch the graph of f. 50. f(x) = ln(x 1) 1Solve each equation for x. 51. (a) e74x=6 (b) ln(3x 10) = 2Solve each equation for x. 52. (a) ln(x2 1) = 3 (b) e2x 3ex + 2 = 0Solve each equation for x. 53. (a) 2x5 = 3 (b) ln x + ln(x 1) = 1Solve each equation for x. 54. (a) ln(ln x) = 1 (b) eax = Cebx, where a bSolve each inequality for x. 55. (a) ln x 0 (b) ex 5Solve each inequality for x. 56. (a) 1 e3x1 2 (b) 1 2 ln x 3(a) Find the domain of f(x) = ln(ex 3). (b) Find f1 and its domain.(a) What are the values of eln 300 and ln(e300)? (b) Use your calculator to evaluate eln 300 and ln(e300). What do you notice? Can you explain why the calculator has trouble?If a bacteria population starts with 100 bacteria and doubles every three hours, then the number of bacteria after t hours is n = f(t) = 1002t/3. (a) Find the inverse of this function and explain its meaning. (b) When will the population reach 50.000?When a camera flash goes off, the batteries immediately begin to recharge the flash's capacitor, which stores electric charge given by Q(t) = Q0(1 e1/a) (The maximum charge capacity is Q0 and t is measured in seconds.) (a) Find the inverse of this function and explain its meaning. (b) How long does it take to recharge the capacitor to 90% of capacity if a = 2?Find the exact value of each expression. 63. (a) cos1 (1) (b) sin1(0.5)Find the exact value of each expression. 64. (a) tan13 (b) arctan(1)65EFind the exact value of each expression. 66. (a) sin1(1/2) (b) cos1(3/2)Find the exact value of each expression. 67. (a) cot1(3) (b) sec1 2Find the exact value of each expression. 68. (a) arcsin(sin(5/4)) (b) cos(2sin1(513))Prove that cos(sin1x)=1x2.Simplify the expression. 70. tan(sin1 x)Simplify the expression. 71. sin(tan1 x)Simplify the expression. 72. sin(2 arccos x)73E74EFind the domain and range of the function g(x) = sin1(3x + 1)(a) Graph the function f(x) = sin(sin1 x) and explain the appearance of the graph. (b) Graph the function g(x) = sin1(sin x). How do you explain the appearance of this graph?(a) If we shift a curve to the left, what happens to its reflection about the line y = x? In view of this geometric principle, find an expression for the inverse of g(x) = f(x + c), where f is a one-to-one function. (b) Find an expression for the inverse of h(x) = f(cx), where c 0.(a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How can you tell whether a given curve is the graph of a function?Discuss four ways of representing a function. Illustrate your discussion with examples.(a) What is an even function? How can you tell if a function is even by looking at its graph? Give three examples of an even function. (b) What is an odd function? How can you tell if a function is odd by looking at its graph? Give three examples of an odd function.What is an increasing function?What is a mathematical model?Give an example of each type of function. (a) Linear function (b) Power function (c) Exponential function (d) Quadratic function (e) Polynomial of degree 5 (f) Rational functionSketch by hand, on the same axes, the graphs of the following functions. (a) f(x) = x (b) g(x) = x2 (c) h(x) = x3 (d) j(x) = x4Draw, by hand, a rough sketch of the graph of each function. (a) y = sin x (b) y = tan x (c) y = ex (d) y = ln x (e) y = 1/x (f) y = |x| (g) y=x (h) y = tan1xSuppose that f has domain A and g has domain B. (a) What is the domain of f + g? (b) What is the domain of fg? (c) What is the domain of f/g?How is the composite function f g defined? What is its domain?Suppose the graph of f is given. Write an equation for each of the graphs that are obtained from the graph of f as follows. (a) Shift 2 units upward. (b) Shift 2 units downward. (c) Shift 2 units to the right. (d) Shift 2 units to the left. (e) Reflect about the x-axis.(a) What is a one-to-one function? How can you tell if a function is one-to-one by looking at its graph? (b) If f is a one-to-one function, how is its inverse function f-1 defined? How do you obtain the graph of f-1 from the graph of f?(a) How is the inverse sine function f(x) = sin1 x defined? What are its domain and range? (b) How is the inverse cosine function f(x) = cos1x defined? What are its domain and range? (c) How is the inverse tangent function f(x) = tan1x defined? What are its domain and range?Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f is a function, then f(s + t) = f(s) + f(t).Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f(s) = f(t), then s = t.Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f is a function, then f(3x) = 3f(x).Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If x1 x2, and f is a decreasing function, then f(x1) f(x2).Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. A vertical line intersects the graph of a function at most once.Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f and g are functions, then f. g = g f.Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f is one-to-one, then f1(x)=1f(x).Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. You can always divide by ex.