Question: The half-life of bismuth-210, 210Bi,

> 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).

> Find the exact value of each expression. a. log1040 + log102.5 b. log860 - log83 - log85

> Find the exact value of each expression. a. log5 1/125 b. ln(1/e2)

> Find the exact value of each expression. a. log232 b. log82

> 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.

> a. How is the logarithmic function y = logbx defined? b. What is the domain of this function? c. What is the range of this function? d. Sketch the general shape of the graph of the function y = logbx if b > 1.

> Let g(x) = 3 1 − x3 . a. Find g-1. How is it related to g? b. Graph g. How do you explain your answer to part (a)?

> Let f(x) = 1 − x2 , 0 ≤ x ≤ 1. a. Find f -1. How is it related to f ? b. Identify the graph of f and explain your answer to part (a).

> Use the given graph off to sketch the graph off -1. 1 2

> Use the given graph off to sketch the graph off -1. yA 1 1

> Find an explicit formula for f -1 and use it to graph f -1, f, and the line y = x on the same screen. To check your work, see whether the graphs of f and f -1 are reflections about the line. f(x) = 1 + e-x

> 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) o

> Find an explicit formula for f -1 and use it to graph f -1, f, and the line y = x on the same screen. To check your work, see whether the graphs of f and f -1 are reflections about the line. f(x) = 4x + 3

> Find a formula for the inverse of the function. y = 1 – e-x / 1 + e-x

> Find a formula for the inverse of the function. y = ln(x + 3)

> Find a formula for the inverse of the function. y = x2 - x, x ≥ 1/2

> Find a formula for the inverse of the function. f(x) = e2x-1

> Find a formula for the inverse of the function. f(x) = 4x – 1/2x + 3

> Find a formula for the inverse of the function. f(x) = 1 + 2 + 3x

> In the theory of relativity, the mass of a particle with speed is 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. mo m = f(v) V1 – v²/c²

> The formula C = 5/9(F – 32), 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?

> The graph off is given. a. Why is f one-to-one? b. What are the domain and range of f -1? c. What is the value of f -1(2)? d. Estimate the value of f -1(0). 1 1

> 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. a. Use the readings to sketch a rough graph of T as a function of t. b. Use yo

> If g(x) = 3 + x + ex, find g-1(4).

> If f(x) = x5 + x3 + x, find f -1(3) and f(f -1(2)).

> Assume that f is a one-to-one function. a. If f(6) = 17, what is f -1(17)? b. If f -1(3) = 2, what is f(2)?

> A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one. f(t) is your height at age t.

> A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one. 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. g(x) = 3

> A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one. g(x) = 1 - sin x

> A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one. f(x) = x4 - 16

> A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one. f(x) = 2x - 3

> A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.

> 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

> 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. у.

> A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.

> Use the Law of Exponents to rewrite and simplify the expression. a. x2n . x3n-1 /xn+2 b. √a

> Use the Law of Exponents to rewrite and simplify the expression. a. b8(2b)4 b. (6y3)4/ 2y5

> Use the Law of Exponents to rewrite and simplify the expression. a. 84/3 b. x(3x2)3

> Use the Law of Exponents to rewrite and simplify the expression. a. 4-3 /2-8 b. 1 / 3 x4

> Graph several members of the family of functions f(x) = 1/ 1 + aebx where a > 0. How does the graph change when b changes? How does it change when a changes?

> If you graph the function you’ll see that f appears to be an odd function. Prove it. 1- e 1/1 f(x) 1+ e ,1/1

> 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

> 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.

> 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.

> After 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 exponentia

> Use the graph of V in Figure 11 to estimate the half-life of the viral load of patient 303 during the first month of treatment. From Figure 11: 60 40+ 20 i (days) 10 20 30 RNA copies / mL

> 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

> 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 func

> A researcher is trying to determine the doubling time for a population of the bacterium Giardia lamblia. He starts a culture in a nutrient solution and estimates the bacteria count every four hours. His data are shown in the table. a. Make a scatter pl

> Use a graph to estimate the values of x such that ex > 1,000,000,000.

> 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 ?

> Compare the functions f(x) = x5 and 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?

> 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 tim

> If is it true that f = g? x² – x S(x)· X - 1 х) — х and

> Suppose the graphs of f(x) = x2 and g(x) = 2x are drawn on a coordinate grid where the unit of measurement is 1 inch. Show that, at a distance 2 ft to the right of the origin, the height of the graph off is 48 ft but the height of the graph of t is about

> Suppose you are offered a job that lasts one month. Which of the following methods of payment do you prefer? I. One million dollars at the end of the month. II. One cent on the first day of the month, two cents on the second day, four cents on the thir

> If f(x) = 5x, show that f(x + h) – f(x) 5* – 1 5* h h

> Find the exponential function f(x) = Cbx whose graph is given. y. (-1, 3) jen

> Find the exponential function f(x) = Cbx whose graph is given. (3, 24), (1, 6)

> Find the domain of each function. a. g(t) = 10t − 100 b. g(t) = sin(et - 1)

> Find the domain of each function. a. f(x) = 1 - ex2 / 1 - e1-x2 b. f(x) = 1 + x / ecos x

> 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.

> 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 to the right. c. reflecting about the x-axis. d. reflecting about the y-axis. e. reflecting about the x-axis and t

> 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. Y = 2(1-ex) From Figure 3: From figure 15: yt 10 4 2 1.5 114 y. y=e m=

> 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.

> 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. y = 1 – 1/2e-x From Figure 3: From figure 15: yt 10

> 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. y = e|x| From Figure 3: From figure 15: yt 10 4 2 1.5 114 y. y=e m=1

> 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. y = -2-x From Figure 3: From figure 15: yt 10 4 2 1.5 114 y. y=e m=1

> 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. y = (0.5)x-1 From Figure 3: From figure 15: yt 10 4 2 1.5 114 y. y=e m

> 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. y = 4x - 1 From Figure 3: From figure 15: yt 10 4 2 1.5 114 y. y=e m=1

> Graph the given functions on a common screen. How are these graphs related? y = 0.9x, y = 0.6x, y = 0.3x, y = 0.1x

> Graph the given functions on a common screen. How are these graphs related? y = 3x, y = 10x, y = (1/3)x, y = (1/10)x

> Graph the given functions on a common screen. How are these graphs related? y = ex, y = e2x, y = 8x, y = 82x

> Graph the given functions on a common screen. How are these graphs related? y = 2x, y = ex, y = 5x, y = 20x

> a. How is the number e defined? b. What is an approximate value for e? c. What is the natural exponential function?

> 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.

> 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 of the graph of the exponential function for each of the fol

> The graph off is given. Draw the graphs of the following functions. a. y = f(x) - 3 b. y = f(x + 1) c. y = 1/2 f(x) d. y = -f (x) 0 1 2.

> The graph of y = f(x) is given. Match each equation with its graph and give reasons for your choices. a. y = f(x - 4) b. y = f(x) + 3 c. y = 1/3 f(x) d. y = -f(x - 4) e. y = 2f(x + 6) 3 -3 3 6 -3 5)

> Explain how each graph is obtained from the graph of y = 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(1/8 x)

> Suppose the graph off is given. Write equations for the graphs that are obtained from the graph off 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-a

> Suppose g is an odd function and let h = f 0 g. Is h always an odd function? What if f is odd? What if f is even?

> Suppose g is an even function and let h = f 0 g. Is h always an even function?

> If f(x) = x + 4 and h(x) = 4x - 1, find a function g such that g 0 f = h.

> a. If g(x) = 2x + 1 and h(x) = 4x2 + 4x + 7, find a function f such that f 0 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 functi

> If you invest x dollars at 4% interest compounded annually, then the amount A(x) of the investment after one year is A(x) = 1.04x. Find A 0 A, A 0 A 0 A, and A 0 A 0 A 0 A. What do these compositions represent? Find a formula for the composition of n cop

> Sketch a rough graph of the outdoor temperature as a function of time during a typical spring day.

> Let f and t be linear functions with equations f(x) = m1x + b1 and g(x) = m2x + b2. Is f 0 t also a linear function? If so, what is the slope of its graph?

> 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 th

> 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 th

> An 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 betwe

> A ship is moving at a speed of 30 km/h parallel to a straight shoreline. The ship is 6 km from shore and it passes a lighthouse at noon. a. Express the distance s between the lighthouse and the ship as a function of d, the distance the ship has traveled

> 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, fi

> 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

> Use the given graphs of f and g to estimate the value of f(g(x)) for x = 25, 24, 23,..., 5. Use these estimates to sketch a rough graph of f 0 g. 1 f

> 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 0 g)(0) d. (g 0 f)(6) e. (g 0 g)(-2) f. (f 0 f)(4) 19 f 2 2.

> Use the table to evaluate each expression. a. f(g(1)) b. g(f(1)) c. f(f(1)) d. g(g(1)) e. (g 0 f)(3) f. (f 0 g)(6) 1 3 4 5 f(x) 3 1 4 2 2 5 g(x) 3 2 1 2 3

> Sketch a rough graph of the number of hours of daylight as a function of the time of year.