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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
If f(x)=x22x, find f(1),f(x2),f(t), and f(p1).State the domain and range of f(x)=(x2+1)1.If f(x)=x2+1 and g(x)=x2, find fg and gf.Refer to Figure 1.12. Find the hiker's average speed during the first mile of the trail and then determine the hikers average speed in the time interval from 3.9 to 4.1 hours.Explain why the graph of a nonzero function is never symmetric with respect to the x-axis.Use the terms domain, range, independent variable, and dependent variable to explain how a function relates one variable to another variable.Is the independent variable of a function associated with the domain or range? Is the dependent variable associated with the domain or range?Vertical line test Decide whether graphs A, B, or both represent functions. 11.The entire graph of f is given. State the domain and range of f.Which statement about a function is true? (i) For each value of x in the domain, there corresponds one unique value of y in the range; (ii) for each value of y in the range, there corresponds one unique value of x in the domain. Explain.Determine the domain and range of g(x)=x21x1. Sketch a graph of g.Determine the domain and range of f(x)=3x210.Domain in context Determine an appropriate domain of each function. Identify the independent and dependent variables. 21. A stone is thrown vertically upward from the ground at a speed of 40 m/s at time t = 0. Its distance d (in meters) above the ground (neglecting air resistance) is approximated by the function f(t) = 40t 5t2.Domain in context Determine an appropriate domain of each function. Identify the independent and dependent variables. 23. A cylindrical water tower with a radius of 10 m and a height of 50 m is filled to a height of h. The volume V of water (in cubic meters) is given by the function g(h) = 100h.If f(x) = 1/(x3 + 1), what is f(2)? What is f(y2)?Let f(x)=2x+1 and g(x)=1/(x1). Simplify the expressions f(g(1/2)),g(f(4)), and g(f(x)).Find functions f and g such that f(g(x))=(x2+1)5. Find a different pair of functions and g that also satisfy f(g(x))=(x2+1)5.Explain how to find the domain of fg if you know the domain and rnge of f and g.If f(x)=x and g(x)=x32, simplify the expressions (fg)(3),(ff)(64),(gf)(x), and (fg)(x).Composite functions from graphs Use the graphs of f and g in the figure to determine the following function values. a. (f g)(2) b. g(f(2)) c. f(g(4)) d. g(f(5)) e. f(f(8)) f. g(f(g(5)))Composite functions from tables Use the table to evaluate the given compositions. a. h(g(0)) b. g(f(4)) c. h(h(0)) d. g(h(f(4))) e. f(f(f(1))) f. h(h(h(0))) g. f(h(g(2))) h. g(f(h(4))) i. g(g(g(1))) j. f(f(h(3)))Rising radiosonde The National Weather Service releases approximately 70,000 radiosondes every year to collect data from the atmosphere. Attached to a balloon, a radiosonde rises at about 1000 ft/mm until the balloon bursts in the upper atmosphere Suppose a radiosonde is released from a point 6 ft above the ground and that 5 seconds later, it is 83 ft above the ground Let f(t) represent the height (in feet) that the radiosonde is above the ground t seconds after it is released. Evaluate f(5)f(0)50 and interpret the meaning of this quotient.World record free fall On October 14, 2012, Felix Baumgartner stepped off a balloon capsule at an altitude of 127,852.4 feet and began his free fall. It is claimed that Felix reached the speed of sound 34 seconds into his fall at an altitude of 109,731 feet and that he continued to fall at supersonic speed for 30 seconds until he was at an altitude of 75,330.4 feet. Let f(t) equal the distance that Felix had fallen t seconds after leaving his capsule. Calculate f(0), f(34), f(64), and his average supersonic speed f(64)f(34)6434 (in ft/s) over the time interval [34, 64] (Source http://www.redbullstratos.com)Suppose f is an even function with f(2) = 2 and g is an odd function with g(2) = 2. Evaluate f(2), g(2), f(g(2)), and g(f(2)).Complete the left half of the graph of g if g is an odd function.21ESymmetry in graphs State whether the functions represented by graphs A, B, and C in the figure are even, odd, or neither.Domain and range State the domain and range of the function. 23.f(x)=x25x+6x2Domain and range State the domain and range of the function. 24.f(x)=x22xDomain and range State the domain and range of the function. 25.f(x)=7x2Domain and range State the domain and range of the function. 26.f(x)=25x2Domain State the domain of the function. 27.h(u)=u13Domain State the domain of the function. 28.F(w)=2w4Domain State the domain of the function. 29.f(x)=(9x2)3/2Domain State the domain of the function. 30.g(t)=11+t2Launching a rocket A small rocket is launched vertically upward from the edge of a cliff 80 ft off the ground at a speed of 96 ft/s. Its height (in feet) above the ground is given by h(t) = 16t2 + 96t + 80, where t represents time measured in seconds. a. Assuming the rocket is launched at t = 0, what is an appropriate domain for h? b. Graph h and determine the time at which the rocket reaches its highest point. What is the height at that time?Draining a tank (Torricellis law) A cylindrical tank with a cross-sectional area of 10 m2 is filled to a depth of 25 m with water. At t = 0 s. a drain in the bottom of the tank with an area of 1 m2 is opened, allowing water to flow out of the tank. The depth of water in the tank (in meters) at time t0 is d(t)=(50.22t)2. a. Check that d(0) = 25. as specified b. At what time is the tank empty? c. What is an appropriate domain for d?Composite functions and notation Let f(x) = x2 4, g(x) = x3, and F(x) = l/(x 3). Simplify or evaluate the following expressions. 27. g(1/z)Composite functions and notation Let f(x) = x2 4, g(x) = x3, and F(x) = l/(x 3). Simplify or evaluate the following expressions. 28. F(y4)Composite functions and notation Let f(x) = x2 4, g(x) = x3, and F(x) = l/(x 3). Simplify or evaluate the following expressions. 29. F(g(y))Composite functions and notation Let f(x) = x2 4, g(x) = x3, and F(x) = l/(x 3). Simplify or evaluate the following expressions. 30. f(g(w))Composite functions and notation Let f(x) = x2 4, g(x) = x3, and F(x) = l/(x 3). Simplify or evaluate the following expressions. 31. g(f(u))Composite functions and notation Let f(x) = x2 4, g(x) = x3, and F(x) = l/(x 3). Simplify or evaluate the following expressions. 32. f(2+h)f(2)hComposite functions and notation Let f(x) = x2 4, g(x) = x3, and F(x) = l/(x 3). Simplify or evaluate the following expressions. 33. F(F(x))Composite functions and notation Let f(x) = x2 4, g(x) = x3, and F(x) = l/(x 3). Simplify or evaluate the following expressions. 33. g(F(f(x)))Composite functions and notation Let f(x) = x2 4, g(x) = x3, and F(x) = l/(x 3). Simplify or evaluate the following expressions. 35. f(x+4)Composite functions and notation Let f(x) = x2 4, g(x) = x3, and F(x) = l/(x 3). Simplify or evaluate the following expressions. 36. F(3x+1x)Working with composite functions Find possible choices for outer and inner functions f and g such that the given function h equals fg. 43.h(x)=(x35)10Working with composite functions Find possible choices for outer and inner functions f and g such that the given function h equals fg. 44.h(x)=2(x6+x2+1)2Working with composite functions Find possible choices for outer and inner functions f and g such that the given function h equals fg. 45.h(x)=x4+2Working with composite functions Find possible choices for outer and inner functions f and g such that the given function h equals fg. 46.h(x)=1x31More composite functions Let f(x) = |x|, g(x) = x2 4, F(x)=x, and G(x) = 1/(x 2). Determine the following composite functions and give their domains. 41. f gMore composite functions Let f(x) = |x|, g(x) = x2 4, F(x)=x, and G(x) = 1/(x 2). Determine the following composite functions and give their domains. 42. g f49EMore composite functions Let f(x) = |x|, g(x) = x2 4, F(x)=x, and G(x) = 1/(x 2). Determine the following composite functions and give their domains. 44. f g GMore composite functions Let f(x) = |x|, g(x) = x2 4, F(x)=x, and G(x) = 1/(x 2). Determine the following composite functions and give their domains. 45. G g fMore composite functions Let f(x) = |x|, g(x) = x2 4, F(x)=x, and G(x)=1/(x2). Determine the following composite functions and give their domains. 52.gFF53EMore composite functions Let f(x) = |x|, g(x) = x2 4, F(x)=x, and G(x) = 1/(x 2). Determine the following composite functions and give their domains. 48. G GMissing piece Let g(x) = x2 + 3. Find a function f that produces the given composition. 49. (f g)(x) = x2Missing piece Let g(x) = x2 + 3. Find a function f that produces the given composition. 50. (fg)(x)=1x2+3Missing piece Let g(x) = x2 + 3. Find a function f that produces the given composition. 51. (f g)(x) = x4 + 6x2 + 9Missing piece Let g(x) = x2 + 3. Find a function f that produces the given composition. 52. (f g)(x) = x4 + 6x2 + 20Missing piece Let g(x) = x2 + 3. Find a function f that produces the given composition. 53. (g f)(x) = x4 + 3Missing piece Let g(x) = x2 + 3. Find a function f that produces the given composition. 54. (g f)(x) = x2/3 + 3Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The range of f(x) = 2x 38 is all real numbers. b. The relation y = x6 + 1 is not a function because y = 2 for both x = 1 and x = 1. c. If f(x) = x1, then f(1/x) = 1/f(x). d. In general, f(f(x)) = (f(x))2. e. In general, f(g(x)) = g(f(x)). f. By definition, f(g(x)) = (f g)(x). g. If f(x) is an even function, then c f(ax) is an even function, where a and c are nonzero real numbers. h. If f(x) is an odd function, then f(x) + d is an odd function, where d is a nonzero real number. i. If f is both even and odd, then f(x) = 0 for all x.Working with difference quotients Simplify the difference quotient f(x+h)f(x)h for the following functions. 62.f(x)=10Working with difference quotients Simplify the difference quotient f(x+h)f(x)h for the following functions. 63.f(x)=3xWorking with difference quotients Simplify the difference quotient f(x+h)f(x)h for the following functions. 58. f(x) = 4x 3Working with difference quotients Simplify the difference quotient f(x+h)f(x)h for the following functions. 57. f(x) = x2Working with difference quotients Simplify the difference quotient f(x+h)f(x)h for the following functions. 60. f(x) = 2x2 3x + 1Working with difference quotients Simplify the difference quotient f(x+h)f(x)h for the following functions. 59. f(x) = 2/xWorking with difference quotients Simplify the difference quotient f(x+h)f(x)h for the following functions. 60. f(x)=xx+1Working with difference quotients Simplify the difference quotient f(x)f(a)xa for the following functions. 69.f(x)=x2+xWorking with difference quotients Simplify the difference quotient f(x)f(a)xa for the following functions. 64. f(x) = 4 4x x2Working with difference quotients Simplify the difference quotient f(x)f(a)xa for the following functions. 63. f(x) = x3 2xWorking with difference quotients Simplify the difference quotient f(x)f(a)xa for the following functions. 62. f(x) = x4Working with difference quotients Simplify the difference quotient f(x)f(a)xa for the following functions. 65. f(x)=4x2Working with difference quotients Simplify the difference quotient f(x)f(a)xa for the following functions. 66. f(x)=1xx2GPS data A GPS device tracks the elevation E (in feet) of a hiker walking in the mountains. The elevation t hours after beginning the hike is given in the figure. a. Find the slope of the secant line that passes through points A and B. Interpret your answer as an average rate of change over the interval 1 t 3 b. Repeat the procedure outlined in part (a) for the secant line that passes through points P and Q. c. Notice that the curve in the figure is horizontal for an interval of time near t = 5.5 hr. Give a plausible explanation for the horizontal line segment.Elevation vs. Distance The following graph, obtained from GPS data, shows the elevation of a hiker as a function of the distance d from the starting point of the trail. a.Find the slope of the secant line that passes through points A and B. Interpret your answer as an average rate of change over the interval 1 d 3. b.Repeat the procedure outlined in part (a) for the secant line that passes through points P and Q c.Notice that the elevation function is nearly constant over the segment of the trail from mile d = 4.5 to mile d = 5. Give a plausible explanation for the horizontal line segment.Interpreting the slope of secant lines In each exercise, a function and an interval of its independent variable are given. The endpoints of the interval are associated with the points P and Q on the graph of the function. a. Sketch a graph of the function and the secant line through P and Q. b. Find the slope of the secant line in part (a), and interpret your answer in terms of an average rate of change over the interval. Include units in your answer. 67. After t seconds, an object dropped from rest falls a distance d = 16t2, where d is measured in feet and 2 t 5.Interpreting the slope of secant lines In each exercise, a function and an interval of its independent variable are given. The endpoints of the interval are associated with the points P and Q on the graph of the function. a. Sketch a graph of the function and the secant line through P and Q. b. Find the slope of the secant line in part (a), and interpret your answer in terms of an average rate of change over the interval. Include units in your answer. 69. The volume V of an ideal gas in cubic centimeters is given by V = 2 / p, where p is the pressure in atmospheres and 0.5 p 2.Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing. 71. f(x) = x4 + 5x2 12Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing. 72. f(x) 3x5 + 2x3 xSymmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing. 73. f(x) x5 x3 2Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing. 74. f(x) = 2|x|83E84ESymmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing. 77. f(x) = x|x|Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing. 78. |x| + |y| = 1Composition of even and odd functions from graphs Assume f is an even function and g is an odd function. Use the (incomplete) graphs of f and g in the figure to determine the following function values. a. f(g(2)) b. g(f(2)) c. f(g(4)) d. g(f(5) 8) e. g(g(7)) f. f(1 f(8))Composition of even and odd functions from tables Assume f is an even function and g is an odd function. Use the (incomplete) table to evaluate the given compositions. a. f(g(1)) b. g(f(4)) c. f(g(3)) d. f(g(2)) e. g(g(1)) f. f(g(0) 1)) g. f(g(g(2))) h. g(f(f(4))) i. g(g(g(1)))Absolute value graph Use the definition of absolute value to graph the equation |x| |y| = 1. Use a graphing utility to check your work.Graphing semicircles Show that the graph of f(x)=10+x2+10x9 is the upper half of a circle. Then determine the domain and range of the function.Graphing semicircles Show that the graph of g(x)=2x2+6x+16 is the lower half of a circle. Then determine the domain and range of the function.Even and odd at the origin a. If f(0) is defined and f is an even function, is it necessarily true that f(0) = 0? Explain. b. If f(0) is defined and f is an odd function, is it necessarily true that f(0) = 0? Explain.Polynomial calculations Find a polynomial f that satisfies the following properties. (Hint: Determine the degree of f; then substitute a polynomial of that degree and solve for its coefficients.) 85. f(f(x)) = 9x 8Polynomial calculations Find a polynomial f that satisfies the following properties. (Hint: Determine the degree of f; then substitute a polynomial of that degree and solve for its coefficients.) 86. (f(x))2 = 9x2 12x + 4Polynomial calculations Find a polynomial f that satisfies the following properties. (Hint: Determine the degree of f; then substitute a polynomial of that degree and solve for its coefficients.) 87. f(f(x)) = x4 12x2 + 30Polynomial calculations Find a polynomial f that satisfies the following properties. (Hint: Determine the degree of f; then substitute a polynomial of that degree and solve for its coefficients.) 88. (f(x))2 = x4 12x2 + 36Difference quotients Simplify the difference quotients f(x+h)f(x)h and f(x)f(a)xa by rationalizing the numerator. 89. f(x)=xDifference quotients Simplify the difference quotients f(x+h)f(x)h and f(x)f(a)xa by rationalizing the numerator. 90. f(x)=12xDifference quotients Simplify the difference quotients f(x+h)f(x)h and f(x)f(a)xa by rationalizing the numerator. 91. f(x)=3xDifference quotients Simplify the difference quotients f(x+h)f(x)h and f(x)f(a)xa by rationalizing the numerator. 92. f(x)=x2+1Combining even and odd functions Let E be an even function and O be an odd function. Determine the symmetry, if any, of the following functions. 95. E + OCombining even and odd functions Let E be an even function and O be an odd function. Determine the symmetry, if any, of the following functions. 96. E OCombining even and odd functions Let E be an even function and O be an odd function. Determine the symmetry, if any, of the following functions. 101. O ECombining even and odd functions Let E be an even function and O be an odd function. Determine the symmetry, if any, of the following functions. 98. E OAre all polynomials rational functions? Are all algebraic functions polynomials?What is the range of f(x) = x7? What is the range of f(x) = x8?What are the domain and range of f(x)=x1/7? What are the domain and range of f(x)=x1/10?How do you modify the graph of f(x)=1/x to produce the graph of g(x)=1/(x+4)?Give four ways that functions may be defined and represented.What is the domain of a polynomial?Graphs of functions Find the linear functions that correspond to the following graphs. 11.Determine the linear function g whose graph is parallel to the line y=2x+1 and passes through the point (5, 0).What is the domain of a rational function?Describe what is meant by a piecewise linear function.Graphs of piecewise functions Write a definition of the functions whose graphs are given. 19.The graph of y=x is shifted 2 units to the right and 3 units up. Write an equation for this transformed graph.How do you obtain the graph of y = f(x + 2) from the graph of y = f(x)?How do you obtain the graph of y = 3f(x) from the graph of y = f(x)?How do you obtain the graph of y = f(3x) from the graph of y = f(x)?How do you obtain the graph of y = 4(x + 3)2 + 6 from the graph of y = x2?Transformations of y = |x| The functions f and g in the figure are obtained by vertical and horizontal shifts and scalings of y = |x|. Find formulas for f and g. Verify your answers with a graphing utility.Transformations Use the graph of f in the figure to plot the following functions. a. y = f(x) b. y = f(x + 2) c. y = f(x 2) d. y = f(2x) e. y = f(x 1) + 2 f. y = 2f(x)Graph of a linear function Find and graph the linear function that passes through the points (1, 3) and (2, 5).Graph of a linear function Find and graph the linear function that passes through the points (2, 3) and (5, 0).Linear function Find the linear function whose graph passes though the point (3, 2) and is parallel to the line y=3x+8.Linear function Find the linear function whose graph passes though the point (1, 4) and is perpendicular to the line y=14x7.Yeast growth Consider a colony of yeast cells that has the shape of a cylinder. As the number of yeast cells increases, the cross-sectional area A (in mm2) of the colony increases but the height of the colony remains constant If the colony starts from a single cell, the number of yeast cells (in millions) is approximated by the linear function N(A) = CsA, where the constant Cs is known as the cell-surface coefficient Use the given Information to determine the cell-surface coefficient for each of the following colonies of yeast cells, and find the number of yeast cells m the colony when the cross-sectional area A reaches 150 mm2. (Source: Letters in Applied Microbiology, 594, 59, 2014) 19.The scientific name of bakers or brewers yeast (used in making bread, wine, and beer) is Saccharomyces cerevisiae. When the cross-sectional area of a colony of this yeast reaches 100 mm2, there are 571 million yeast cells.Yeast growth Consider a colony of yeast cells that has the shape of a cylinder. As the number of yeast cells increases, the cross-sectional area A (in mm2) of the colony increases but the height of the colony remains constant If the colony starts from a single cell, the number of yeast cells (in millions) is approximated by the linear function N(A) = CsA, where the constant Cs is known as the cell-surface coefficient Use the given Information to determine the cell-surface coefficient for each of the following colonies of yeast cells, and find the number of yeast cells m the colony when the cross-sectional area A reaches 150 mm2. (Source: Letters in Applied Microbiology, 594, 59, 2014) 20.The yeast Rhodotorula glutinis is a laboratory contaminant. When the cross-sectional area of a colony reaches 100 mm2, there are 226 million yeast cells.Demand function Sales records indicate that if Blu-ray players are priced at 250, then a large store sells an average of 12 units per day. If they are priced at 200, then the store sells an average of 15 units per day. Find and graph the linear demand function for Blu-ray sales. For what prices is the demand function defined?Fundraiser The Biology Club plans to have a fundraiser for which 8 tickets will be sold. The cost of room rental and refreshments is 175. Find and graph the function p = f(n) that gives the profit from the fundraiser when n tickets are sold. Notice that f(0) = 175; that is, the cost of room rental and refreshments must be paid regardless of how many tickets are sold. How many tickets must be sold to break even (zero profit)?Bald eagle population Since DDT was banned and the Endangered Species Act was passed in 1973, the number of bald eagles in the United States has increased dramatically (see figure). In the lower 48 states, the number of breeding pairs of bald eagles increased at a nearly linear rate from 1875 pairs in 1986 to 6471 pairs in 2000. a. Use the data points for 1986 and 2000 to find a linear function p that models the number of breeding pairs from 1986 to 2000 (0 t 14). b. Using the function in part (a), approximately how many breeding pairs were in the lower 48 states in 1995? (Source: U.S. Fish and Wildlife Service)Taxicab fees A taxicab ride costs 3.50 plus 2.50 per mile. Let m be the distance (in miles) from the airport to a hotel. Find and graph the function c(m) that represents the cost of taking a taxi from the airport to the hotel. Also determine how much it costs if the hotel is 9 miles from the airport.Defining piecewise functions Write a definition of the function whose graph is given. 25.Graphs of piecewise functions Write a definition of the functions whose graphs are given. 20.Parking fees Suppose that it costs 5 per minute to park at the airport with the rate dropping to 3 per minute after 9 P.M. Find and graph the cost function c(t) for values of t satisfying 0 t 120. Assume that t is the number of minutes after 8 P.M.Taxicab fees A taxicab ride costs 3.50 plus 2.50 per mile for the first 5 miles, with the rate dropping to 1.50 per mile after the fifth mile. Let m be the distance (in miles) from the airport to a hotel. Find and graph the piecewise linear function c(m) that represents the cost of taking a taxi from the airport to a hotel m miles away.Piecewise linear functions Graph the following functions. 23. f(x)={x2xx1ifx12ifx=1Piecewise linear functions Graph the following functions. 24. f(x)={x2x2x2ifx24ifx=2Piecewise linear functions Graph the following functions. 25. f(x)={3x1ifx02x+1ifx0Piecewise linear functions Graph the following functions. 26. f(x)={3x1ifx1x+1ifx1Piecewise linear functions Graph the following functions. 27. f(x)={2x1ifx11if1x12x1ifx1Piecewise linear functions Graph the following functions. 28. f(x)={2x+2ifx0x+2if0x23x/2ifx2Graphs of functions a. Use a graphing utility to produce a graph of the given function. Experiment with different windows to see how the graph changes on different scales. Sketch an accurate graph by hand after using the graphing utility. b. Give the domain of the function. c. Discuss interesting features of the function, such as peaks, valleys, and intercepts (as in Example 5). 29. f(x) = x3 2x2 + 6Graphs of functions a. Use a graphing utility to produce a graph of the given function. Experiment with different windows to see how the graph changes on different scales. Sketch an accurate graph by hand after using the graphing utility. b. Give the domain of the function. c. Discuss interesting features of the function, such as peaks, valleys, and intercepts (as in Example 5). 30. f(x)=2x283Graphs of functions a. Use a graphing utility to produce a graph of the given function. Experiment with different windows to see how the graph changes on different scales. Sketch an accurate graph by hand after using the graphing utility. b. Give the domain of the function. c. Discuss interesting features of the function, such as peaks, valleys, and intercepts (as in Example 5). 31. g(x)=|x24x+3|Graphs of functions a. Use a graphing utility to produce a graph of the given function. Experiment with different windows to see how the graph changes on different scales. Sketch an accurate graph by hand after using the graphing utility. b. Give the domain of the function. c. Discuss interesting features of the function, such as peaks, valleys, and intercepts (as in Example 5). 32. f(x)=3x212x+139EGraphs of functions a. Use a graphing utility to produce a graph of the given function. Experiment with different windows to see how the graph changes on different scales. Sketch an accurate graph by hand after using the graphing utility. b. Give the domain of the function. c. Discuss interesting features of the function, such as peaks, valleys, and intercepts (as in Example 5). 34. f(x)={x1x1ifx10ifx=1Features of a graph Consider the graph of the function f shown in the figure. Answer the following questions by referring to the points AI. a. Which points correspond to the roots (zeros) of f? b. Which points on the graph correspond to high points or peaks (soon to be called local maximum values of f)? c. Which points on the graph correspond to low points or valleys (soon to be called local minimum values of f)? d. As you move along the curve in the positive x-direction, at which point is the graph rising most rapidly? e. As you move along the curve in the positive x-direction, at which point is the graph falling most rapidly?Features of a graph Consider the graph of the function g shown in the figure. a. Give the approximate roots (zeros) of g. b. Give the approximate coordinates of the high points or peaks (soon to be called local maximum values of f). c. Give the approximate coordinates of the low points or valleys (soon to be called local minimum values of f). d. Imagine moving along the curve in the positive x-direction on the interval [0, 3]. Give the approximate coordinates of the point at which the graph is rising most rapidly. e. Imagine moving along the curve in the positive x-direction on the interval [0, 3]. Give the approximate coordinates of the point at which the graph is falling most rapidly.Relative acuity of the human eye The fovea centralis (or fovea) is responsible for the sharp central vision that humans use for reading and other detail-oriented eyesight. The relative acuity of a human eye, which measures the sharpness of vision, is modeled by the function R()=0.5680.331+0.568 where (in degrees) is the angular deviation of the line of sight from the center of the fovea (see figure). a. Graph R, for 15 15. b. For what value of is R maximized? What does this fact indicate about our eyesight? c. For what values of do we maintain at least 90% of our maximum relative acuity? (Source: The Journal of Experimental Biology, 203, Dec 2000)Slope functions Determine the slope function S(x) for the following functions. 44.f(x=3)Slope functions Determine the slope function for the following functions. 35. f(x) = 2x + lSlope functions Determine the slope function for the following functions. 36. f(x) = |x|Slope functions Determine the slope function S(x) for the following functions. 47.Use the figure for Exercise 7Slope functions Determine the slope function S(x) for the following functions. 48.Use the figure for Exercise 26Area functions Let A(x) be the area of the region bounded by the t-axis and the graph of y = f(t) from t = 0 to t = x. Consider the following functions and graphs. a. Find A(2). b. Find A(6). c. Find a formula for A(x). 39. f(t) = 6Area functions Let A(x) be the area of the region bounded by the t-axis and the graph of y = f(t) from t = 0 to t = x. Consider the following functions and graphs. a. Find A(2). b. Find A(6). c. Find a formula for A(x). 40. f(t)=t2Area functions Let A(x) be the area of the region bounded by the t-axis and the graph of y = f(t) from t = 0 to t = x. Consider the following functions and graphs. a. Find A(2). b. Find A(6). c. Find a formula for A(x). 41. f(t){2t+8ift32ift3Area functions Let A(x) be the area of the region bounded by the t-axis and the graph of y = f(t) from t = 0 to t = x. Consider the following functions and graphs. a. Find A(2). b. Find A(6). c. Find a formula for A(x). 42. f(t) = |t 2| + 1Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. All polynomials are rational functions, but not all rational functions are polynomials. b. If f is a linear polynomial, then f f is a quadratic polynomial. c. If f and g are polynomials, then the degrees of f g and g f are equal. d. To graph g(x) = f(x + 2), shift the graph of f 2 units to the right.54ETransformations of f(x) = x2 Use shifts and scalings to transform the graph of f(x) = x2 into the graph of g. Use a graphing utility to check your work. a. g(x) = f(x 3) b. g(x) = f(2x 4) c. g(x) = 3f(x 2) + 4 d. g(x)=6f(x23)+1Transformations of f(x)=x Use shifts and scalings to transform the graph of f(x)=x into the graph of g. Use a graphing utility to check your work. a. g(x) = f(x + 4) b. g(x) = 2f(2x 1) c. g(x)=x1 d. g(x)=3x15Shifting and scaling Use shifts and scalings to graph the given functions. Then check your work with a graphing utility. Be sure to identify an original function on which the shifts and scalings are performed. 47. f(x) = (x 2)2 + 1Shifting and scaling Use shifts and scalings to graph the given functions. Then check your work with a graphing utility. Be sure to identify an original function on which the shifts and scalings are performed. 48. f(x) = x2 2x + 3 (Hint: Complete the square first.)Shifting and scaling Use shifts and scalings to graph the given functions. Then check your work with a graphing utility. Be sure to identify an original function on which the shifts and scalings are performed. 49. g(x) = 3x2Shifting and scaling Use shifts and scalings to graph the given functions. Then check your work with a graphing utility. Be sure to identify an original function on which the shifts and scalings are performed. 50. g(x) = 2x3 161EShifting and scaling Use shifts and scalings to graph the given functions. Then check your work with a graphing utility. Be sure to identify an original function on which the shifts and scalings are performed. 52. p(x) = x2 + 3x 563EShifting and scaling Use shifts and scalings to graph the given functions. Then check your work with a graphing utility. Be sure to identify an original function on which the shifts and scalings are performed. 54. h(x) = |3x 6| + 1Intersection problems Find the following points of intersection. 65.The point(s) of intersection of the curves y=42x and y=2x2Intersection problems Use analytical methods to find the following points of intersection. Use a graphing utility to check your work. 56. Find the point(s) of intersection of the parabola y = x2 + 2 and the line y = x + 4.Intersection problems Use analytical methods to find the following points of intersection. Use a graphing utility to check your work. 57. Find the point(s) of intersection of the parabolas y = x2 and y = x2 + 8x.Two semicircles The entire graph of f consists of the upper half of a circle of radius 2 centered at the origin and the lower half of a circle of radius 3 centered at (5, 0). Find a piecewise function for f and plot a graph of f.Piecewise function Plot a graph of the function f(x)={32xif0x23+x2if2x625(x6)2if6x11.70E71E72E73E74E75E76ETennis probabilities Suppose the probability of a server winning any given point in a tennis match is a constant p, with 0 p 1. Then the probability of the server winning a game when serving from deuce is f(p)=p212p(1p). a. Evaluate f(0.75) and interpret the result. b. Evaluate f(0.25) and interpret the result. (Source: The College Mathematics Journal 38, 1, Jan 2007).Temperature scales a. Find the linear function C = f(F) that gives the reading on the Celsius temperature scale corresponding to a reading on the Fahrenheit scale. Use the facts that C = 0 when F = 32 (freezing point) and C = 100 when F = 212 (boiling point). b. At what temperature are the Celsius and Fahrenheit readings equal?Automobile lease vs. purchase A car dealer offers a purchase option and a lease option on all new cars. Suppose you are interested in a car that can be bought outright for 25,000 or leased for a start-up fee of 1200 plus monthly payments of 350. a. Find the linear function y = f(m) that gives the total amount you have paid on the lease option after m months. b. With the lease option, after a 48-month (4-year) term, the car has a residual value of 10,000, which is the amount that you could pay to purchase the car. Assuming no other costs, should you lease or buy?Walking and rowing Kelly has finished a picnic on an island that is 200 m off shore (see figure). She wants to return to a beach house that is 600 m from the point P on the shore closest to island. She plans to row a boat to a point on shore x meters from P and then jog along the (straight) shore to the house. a. Let d(x) be the total length of her trip as a function of x. Find and graph this function. b. Suppose that Kelly can row at 2 m/s and jog at 4 m/s. Let T(x) be the total time for her trip as a function of x. Find and graph y = T(x). c. Based on your graph in part (b), estimate the point on the shore at which Kelly should land to minimize the total time of her trip. What is that minimum time?Optimal boxes Imagine a lidless box with height h and a square base whose sides have length x. The box must have a volume of 125 ft3. a. Find and graph the function S(x) that gives the surface area of the box, for all values of x 0. b. Based on your graph in part (a), estimate the value of x that produces the box with a minimum surface area.Composition of polynomials Let f be an nth-degree polynomial and let g be an mth-degree polynomial. What is the degree of the following polynomials? a. ff b. ff c. fg d. fgParabola vertex property Prove that if a parabola crosses the x-axis twice, the x-coordinate of the vertex of the parabola is halfway between the x-interceptsParabola properties Consider the general quadratic function f(x) = ax2 + bx + c, with a 0. a. Find the coordinates of the vertex in terms of a, b, and c. b. Find the conditions on a, b, and c that guarantee that the graph of f crosses the x-axis twice.Factorial function The factorial function is defined for positive integers as n!=n(n1)(n2)321. a. Make a table of the factorial function, for n = 1, 2, 3, 4, 5. b. Graph these data points and then connect them with a smooth curve. c. What is the least value of n for which n! 106?Is it possible to raise a positive number b to a power and obtain a negative number? Is it possible to obtain zero?Explain why f(x)=(13)x is a decreasing function.What is the inverse of f(x)=13x? What is the inverse of f(x)=x7?The function that gives degrees Fahrenheit in terms of degrees Celsius is F = 9C/5 + 32. Why does this function have an inverse?On what interval(s) does the function f(x) = x3 have an inverse?What is the domain of f(x)=logbx2? What is the range of f(x)=logbx2?For b 0, what are the domain and range of f(x) = bx?Give an example of a function that is one-to-one on the entire real number line.Sketch a graph of a function that is one-to-one on the interval (, 0 ] but is not one-to-one on (, ).Sketch a graph of a function that is one-to-one on the intervals (, 2], and [2, ) but is not one-to-one on (, ).One-to-one functions 11. Find three intervals on which f is one-to-one, making each interval as large as possible.Find four intervals on which f is one-to-one, making each interval as large as possible.Explain why a function that is not one-to-one on an interval I cannot have an inverse function on I.Use the graph of f to find f1(2),f1(9), and f1(12).Find the inverse of the function f(x) = 2x. Verify that f(f1(x)=x) and f1(f(x))=x.Find the inverse of the function f(x)=x, for x 0. Verify that f(f1(x))=x and f1(f(x))=x.Graphs of inverses Sketch the graph of the inverse function. 39.Graphs of inverses Sketch the graph of the inverse function. 40.The parabola y=x2+1 consists of two one-to-one functions, g1(x) and g2(x). Complete each exercise and confirm that your answers are consistent with the graphs displayed in the figure. 13.Find formulas for g1(x) and g11(x). State the domain and range of each function.The parabola y=x2+1 consists of two one-to-one functions, g1(x) and g2(x). Complete each exercise and confirm that your answers are consistent with the graphs displayed in the figure. 14.Find formulas for g2(x) and g21(x). State the domain and range of each function.Explain the meaning of logbx.How is the property bx+ y = bxby related to the property logb (xy) = logb x + logb y?For b 0 with b 1, what are the domain and range of f(x) = logb x and why?Express 25 using base e.Evaluate each expression without a calculator. a. log101000 b. log216 c. log100.01 d. ln e3 e. ln eFor a certain constant a 1, ln a 3.8067. Find approximate values of log2a and loga2 using the fact that ln 2 0.6931.Where do inverses exist? Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse. 15. f(x) = 3x + 4Where do inverses exist? Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse. 16. f(x) = |2x + 1|Where do inverses exist? Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse. 17. f(x) = l/(x 5)Where do inverses exist? Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse. 18. f(x) = (6 x)2Where do inverses exist? Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse. 19. f(x) = 1/x2Where do inverses exist? Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse. 20. f(x) = x2 2x + 8 (Hint: Complete the square.)Graphing inverse functions Find the inverse function (on the given interval, if specified) and graph both f and f1 on the same set of axes. Check your work by looking for the required symmetry in the graphs. 31. f(x) = 8 4xGraphing inverse functions Find the inverse function (on the given interval, if specified) and graph both f and f1 on the same set of axes. Check your work by looking for the required symmetry in the graphs. 28.f(x)=3x+5Graphing inverse functions Find the inverse function (on the given interval, if specified) and graph both f and f1 on the same set of axes. Check your work by looking for the required symmetry in the graphs. 29.f(x)=x+2,forx2Graphing inverse functions Find the inverse function (on the given interval, if specified) and graph both f and f1 on the same set of axes. Check your work by looking for the required symmetry in the graphs. 30.f(x)=3x,forx3Graphing inverse functions Find the inverse function (on the given interval, if specified) and graph both f and f1 on the same set of axes. Check your work by looking for the required symmetry in the graphs. 31.f(x)=(x2)21,forx2Graphing inverse functions Find the inverse function (on the given interval, if specified) and graph both f and f1 on the same set of axes. Check your work by looking for the required symmetry in the graphs. 32.f(x)=x2+4,forx0Finding inverse functions Find the inverse f1(x) of each function (on the given interval, if specified). 33.f(x)=2/(x2+1),forx0Finding inverse functions Find the inverse f1(x) of each function (on the given interval, if specified). 34.f(x)=6/(x29),forx3Finding inverse functions Find the inverse f1(x) of each function (on the given interval, if specified). 35.f(x)=e2x+6Finding inverse functions Find the inverse f1(x) of each function (on the given interval, if specified). 36.f(x)=4e5xFinding inverse functions Find the inverse f1(x) of each function (on the given interval, if specified). 37.f(x)=ln(3x+1)Finding inverse functions Find the inverse f1(x) of each function (on the given interval, if specified). 38.f(x)=log104xFinding inverse functions Find the inverse f1(x) of each function (on the given interval, if specified). 39.f(x)=102xFinding inverse functions Find the inverse f1(x) of each function (on the given interval, if specified). 40.f(x)=1/(ex+1)Finding inverse functions Find the inverse f1(x) of each function (on the given interval, if specified). 41.f(x)=ex/(ex+2)Finding inverse functions Find the inverse f1(x) of each function (on the given interval, if specified). 42.f(x)=x/(x2),forx2Splitting up curves The unit circle x2 + y2 = 1 consists of four one-to-one functions, f1(x), f2(x), f3(x) and f4(x) (see figure). a. Find the domain and a formula for each function. b. Find the inverse of each function and write it as y = f1(x).Splitting up curves The equation y4 = 4x2 is associated with four one-to-one functions f1(x), f2(x), f3(x) and f4(x) (see figure). a. Find the domain and a formula for each function. b. Find the inverse of each function and write it as y = f1(x).Properties of logarithms Assume logb x = 0.36, logb y = 0.56, and logb z = 0.83. Evaluate the following expressions. 47. logbxyProperties of logarithms Assume logb x = 0.36, logb y = 0.56, and logb z = 0.83. Evaluate the following expressions. 48. logb x2Properties of logarithms Assume logb x = 0.36, logb y = 0.56, and logb z = 0.83. Evaluate the following expressions. 49. logb x2Properties of logarithms Assume logb x = 0.36, logb y = 0.56, and logb z = 0.83. Evaluate the following expressions. 50. logbxyzProperties of logarithms Assume logb x = 0.36, logb y = 0.56, and logb z = 0.83. Evaluate the following expressions. 51. logbxz3Properties of logarithms Assume logb x = 0.36, logb y = 0.56, and logb z = 0.83. Evaluate the following expressions. 52. logbb2x5/2ySolving logarithmic equations Solve the following equations. 41. log10 x = 3Solving logarithmic equations Solve the following equations. 42. log5 x = 1Solving logarithmic equations Solve the following equations. 43. loggx=13Solving logarithmic equations Solve the following equations. 44. logb 125 = 3Solving logarithmic equations Solve the following equations. 45. ln x = 1Solving logarithmic equations Solve the following equations. 46. ln y = 3Solving equations Solve the following equations. 53. 7x = 21Solving equations Solve the following equations. 54. 2x = 55Solving equations Solve the following equations. 55. 33x4 = 15Solving equations Solve the following equations. 56. 53x = 29Using inverse relations One hundred grams of a particular radioactive substance decays according to the function m(t) = 100 et/650, where t 0 measures time in years. When does the mass reach 50 grams?Mass of juvenile desert tortoises In a study conducted at the University of New Mexico, it was found that the mass m(t) (in grams) of juvenile desert tortoises t days after a switch to a particular diet is accurately described by the function m(t)=m0e0.004t, where m0 is the mass of the tortoise at the time of the diet switch. According to this function, how long does it take a juvenile desert tortoise to reach a mass of 150 g if it had a mass of 64 g when its diet was switched? (Source: Physiological and Biochemical Zoology, 85, 1, 2012)Investment Problems An investment of P dollars is deposited in a savings account that is compounded monthly with an annual interest rate of r, where r is expressed as a decimal. The amount of money A in the account after t years is given by A=P(1+r/12)12t. Use this equation to complete the following exercises. 63.Determine the time it takes an investment of 1000 to increase to 1100 dollars if it is placed in an account that is compounded monthly with an annual interest rate of 1%(r = 0.01).Investment Problems An investment of P dollars is deposited in a savings account that is compounded monthly with an annual interest rate of r, where r is expressed as a decimal. The amount of money A in the account after t years is given by A=P(1+r/12)12t. Use this equation to complete the following exercises. 64.Determine the time it takes an investment of 20,000 to increase to 22,000 if it is placed in an account that is compounded monthly with an annual interest rate of 2.5%.Height and time The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft/s is given by h = f(t) = 64t 16t2, where t is measured in seconds after the hit a. Is this function one-to-one on the interval 0 t 4? b. Find the inverse function that gives the time t at which the ball is at height h as the ball travels upward. Express your answer in the form t = f1 (h). c. Find the inverse function that gives the time t at which the ball is at height h as the ball travels downward. Express your answer in the form t = f1 (h). d. At what time is the ball at a height of 30 ft on the way up? e. At what time is the ball at a height of 10 ft on the way down?Velocity of a skydiver The velocity of a skydiver (in m/s) t seconds after jumping from a plane is v(t) = 600(1 ekt/60)/k, where k 0 is a constant. The terminal velocity of the skydiver is the value that v(t) approaches as t becomes large. Graph v with k = 11 and estimate the terminal velocity.Calculator base change Write the following logarithms in terms of the natural logarithm. Then use a calculator to find the value of the logarithm, rounding your result to four decimal places. 59. log2 15Calculator base change Write the following logarithms in terms of the natural logarithm. Then use a calculator to find the value of the logarithm, rounding your result to four decimal places. 60. log3 30Calculator base change Write the following logarithms in terms of the natural logarithm. Then use a calculator to find the value of the logarithm, rounding your result to four decimal places. 61. log4 40Calculator base change Write the following logarithms in terms of the natural logarithm. Then use a calculator to find the value of the logarithm, rounding your result to four decimal places. 62. log6 60Changing bases Convert the following expressions to the indicated base. 63. 2x using base eChanging bases Convert the following expressions to the indicated base. 64. 3sin x using base eChanging bases Convert the following expressions to the indicated base. 65. In |x| using base 5Changing bases Convert the following expressions to the indicated base. 66. log2 (x2 + 1) using base eChanging bases Convert the following expressions to the indicated base. 67. a1/ln a using base e, for a 0 and a 1Changing bases Convert the following expressions to the indicated base. 68. a1/log10a using base 10, for a 0 and a 1Explain why or why not Determine whether the following statements are true and give an explanation counterexample. a. If y = 3x, then x=y3 b. logbxlogby=logbxlogby c. log5 46 = 4 log5 6 d. 2=10log102 e. 2 = ln 2e f. If f(x) = x2 + 1, then f1(x) = 1/(x2 + 1). g. If f(x) = 1/x, then f1(x) = 1/x.Graphs of exponential functions The following figure shows the graphs of y = 2x, y = 3x, y = 2x, and y = 3x. Match each curve with the correct function.Graphs of logarithmic functions The following figure shows the graphs of y = log2 x, y = log4 x, and y = log10 x. Match each curve with the correct function.Graphs of modified exponential functions Without using a graphing utility, sketch the graph of y = 2x. Then on the same set of axes, sketch the graphs of y = 2x, y = 2x1, y = 2x + 1, and y = 22x.Graphs of modified logarithmic functions Without using a graphing utility, sketch the graph of y = log2 x. Then on the same set of axes, sketch the graphs of y = log2 (x 1), y = log2 x2, y = (log2 x)2, and y = log2 x + 1.Population model A culture of bacteria has a population of 150 cells when it is first observed. The population doubles every 12 hr, which means its population is governed by the function p(t) = 150 2t/12, where t is the number of hours after the first observation. a. Verify that p(0) = 150, as claimed. b. Show that the population doubles every 12 hr, as claimed. c. What is the population 4 days after the first observation? d. How long does it take the population to triple in size? e. How long does it take the population to reach 10,000?Charging a capacitor A capacitor is a device that stores electrical charge. The charge on a capacitor accumulates according to the function Q(t) = a(1 et/c), where t is measured in seconds, and a and c 0 are physical constants. The steady-state charge is the value that Q(t) approaches as t becomes large. a. Graph the charge function for t 0 using a = 1 and c = 10. Find a graphing window that shows the full range of the function. b. Vary the value of a while holding c fixed. Describe the effect on the curve. How does the steady-state charge vary with a? c. Vary the value of c while holding a fixed. Describe the effect on the curve. How does the steady-state charge vary with c? d. Find a formula that gives the steady-state charge in terms of a and c.Large intersection point Use any means to approximate the intersection point(s) of the graphs of f(x) = ex and g(x) = x123. (Hint: Consider using logarithms.)Finding all inverses Find all the inverses associated with the following functions, and state their domains. 85.f(x)=x22x+6 (Hint: Complete the square first.)Finding all inverses Find all the inverses associated with the following functions, and state their domains. 86.f(x)=x24x3 (Hint: Complete the square first.)Finding all inverses Find all the inverses associated with the following functions and state their domains. 75. f(x) = (x + 1)388EFinding all inverses Find all the inverses associated with the following functions and state their domains. 77. f(x) = 2/(x2 + 2)Finding all inverses Find all the inverses associated with the following functions and state their domains. 78. f(x) = 2x/(x + 2)91E92E93E94E95EInverse of composite functions a. Let g(x) = 2x + 3 and h(x) = x3. Consider the composite function f(x) = g(h(x)). Find f1 directly and then express it in terms of g1 and h1. b. Let g(x) = x2 + 1 and h(x)=x. Consider the composite function f(x) = g(h(x)). Find f1 directly and then express it in terms of g1 and h1. c. Explain why if g and h are one-to-one, the inverse of f(x) = g(h(x)) exists.