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Discovering Advanced Algebra An Investigative Approach 1st edition Jerald Murdock, Ellen Kamischke, Eric Kamischke - Solutions

Given the functions f (x) = - x2 + 2x + 3 and g(x) = (x - 2)2, find these values. a. f (g(3)) b. f (g(2)) c. g( f (0.5)) d. g( f (1)) e. f (g(x)). Simplify to remove all parentheses. f. g( f (x)). Simplify to remove all parentheses.
Aaron and Davis need to write the equation that will produce the graph at right.Aaron: "This is impossible! How are we supposed to know if the parent function is a parabola or a semicircle? If we don't know the parent function, there is no way to write the equation."Davis: "Don't panic yet. I am
Jen and Priya decide to go out to the Hamburger Shack for lunch. They each have a 50-cent coupon from the Sunday newspaper for the Super-Duper-Deluxe $5.49 Value Meal. In addition, if they show their I.D. cards, they'll also get a 10% discount. Jen's server rang up the order as Value Meal, coupon,
Bonnie and Mike are working on a physics project. They need to determine the ohm rating (resistance in ohms) of a resistor. Electrical resistance, measured in ohms, is defined as potential difference, measured in volts, divided by current, measured in amperes (amps). In their project they set up
Begin with the equation of the unit circle, x2 + y2 = 1. a. Apply a horizontal stretch by a factor of 3 and a vertical stretch by a factor of 3, and write the equation that results. b. Sketch the graph. Label the intercepts.
Imagine translating the graph of f(x) = x2 left 3 units and up 5 units, and call the image g(x). a. Give the equation for g(x). b. What is the vertex of the graph of y = g(x)? c. Give the coordinates of the image point that is 2 units to the right of the vertex.
The functions f and g are defined by these sets of input and output values. g = {(1, 2), (- 2, 4), (5, 5), (6, - 2)} f = {(0, - 2), (4, 1), (3, 5), (5, 0)} a. Find g( f (4)). b. Find f (g(- 2)). c. Find f (g( f (3))).
Graph A shows a swimmer's speed as a function of time. Graph B shows the swimmer's oxygen consumption as a function of her speed. Time is measured in seconds, speed in meters per second, and oxygen consumption in liters per minute. Use the graphs to estimate the values.a. The swimmer's speed after
Identify each equation as a composition of functions, a product of functions, or neither. If it is a composition or a product, then identify the two functions that combine to create the equation. a. y = 5 √3 + 2x b. y = 3 + ( | x + 5 | - 3)2 c. y = (x - 5)2 (2 - )
Consider the graph at right.a. Write an equation for this graph. b. Write two functions, f and g, such that the figure is the graph of y = f (g(x)).
The functions f and g are defined by these sets of input and output values. g = {(1, 2), (- 2, 4), (5, 5), (6, - 2)} f = {(2, 1), (4, - 2), (5, 5), (- 2, 6)} a. Find g( f (2)). b. Find f (g(6)). c. Select any number from the domain of either g or f, and find f (g(x)) or g( f (x)),
A, B, and C are gauges with different linear measurement scales. When A measures 12, B measures 13, and when A measures 36, B measures 29. When B measures 20, C measures 57, and when B measures 32, C measures 84.a. Sketch separate graphs for readings of B as a function of A and readings of C as a
The graph of the function y = g(x) is shown at right. Draw a graph of each of these related functions.a. y = ˆšg(x) b. y = | g(x) | c. y = (g(x))2
The two lines pictured at right are f (x) = 2x - 1 and g(x) = 1/2 x + 1/2. Solve each problem both graphically and numerically.a. Find g( f (2)).b. Find f (g(- 1)).c. Pick your own x-value in the domain of f, and find g( f (x)).d. Pick your own x-value in the domain of g, and find f (g(x)).e.
Sketch a graph that shows the relationship between the time in seconds after you start microwaving a bag of popcorn and the number of pops per second. Describe in words what your graph shows.
Use these three functions to find each value: f (x) = -2x + 7 g(x) = x2 - 2 h(x) = (x + 1)2 a. f(4) b. g(-3) c. h(x + 2) - 3 d. f (g(3)) e. g(h(-2)) f. h( f (-1)) g. f (g(a)) h. g( f (a)) i. h( f (a))
The graph of y = f (x) is shown at right. Sketch the graph of each of these functions:a. y = f (x) - 3 b. y = f (x - 3) c. y = 3f (x) d. y = f (-x)
Assume you know the graph of y = f (x). Describe the transformations, in order, that would give you the graph of these functions:a. y = f (x + 2) - 3b.c.
The graph of y = f (x) is shown at right. Use what you know about transformations to sketch these related functions:a. y - 1 = f (x - 2) b. c. y = f (- x) + 1 d. e. y = - f (x - 3) + 1 f.
For each graph, name the parent function and write an equation of the graph.a.b.c.d.e.f.g.h.
The Acme Bus Company has a daily ridership of 18,000 passengers and charges $1.00 per ride. The company wants to raise the fare yet keep its revenue as large as possible. (The revenue is found by multiplying the number of passengers by the fare charged.) From previous fare increases, the company
Evaluate each function at the given value. a. f (x) = 4.753(0.9421)x, x = 5 b. g(h) = 238(1.37)h, h = 14 c. h(t) = 47.3(0.835)t + 22.3, t = 24 d. j(x) = 225(1.0825)x - 3, x = 37?
Each of the red curves is a transformation of the graph of y = 0.5x, shown in black. Write an equation for each red curve.a.b. c. d.
The general form of an exponential equation, y = abx, is convenient when you know the y-intercept. Start with f(0) = 30 and f(1) = 27. This painting, Inspiration Point (2000), by contemporary American artist Nina Bovasso, implies an explosion of exponential growth. a. Find the common ratio. b.
The graph shows a line and the graph of y = f (x). a. Complete the missing values to make a true statement. f (?) = ? . b. Find the equation of the pictured line.
Janell starts 10 m from a motion sensor and walks at 2 m/s toward the sensor. When she is 3 m from the sensor, she stantly turns around and walks at the same speed back to the starting point. a. Sketch a graph of the function that models Janell's walk. b. Give the domain and range of the
Make up two linear functions, f and g. Enter Y1 = f (x) and Y2 = g(x). Enter f (g(x)) in Y3 as Y1(Y2) and g( f (x)) in Y4 as Y2(Y1). a. Display the graphs of f (g(x)) and g(f(x)). Describe the relationship between them. b. Change f (x) or g(x) or both, and again graph f (g(x)) and g(f(x)). Does the
Use geometry software to construct a circle. Label it circle M and measure its area. a. Construct another circle with twice the area of circle M and label it circle L. b. Construct another circle with half the area of circle M and label it circle S. c. Describe the method you used to determine the
You can use different techniques to find the product of two binomials, such as (x - 4)(x + 6). a. Use a rectangle diagram to find the product. b. You can use the distributive property to rewrite the expression (x - 4)(x + 6) as x(x + 6) - 4(x + 6). Use the distributive property again to find all
Record three terms of the sequence, and then write an explicit function for the sequence. a. u0 = 16 un = 0.75un-1 where n (1 b. u0 = 24 un = 1.5un-1 where n ( 1
Evaluate each function at x = 0, x = 1, and x = 2, and then write a recursive formula for the pattern. a. f (x) = 125(0.6)x b. f (x) = 3(2)x
Calculate the ratio of the second term to the first term, and express the answer as a decimal value. State the percent increase or decrease. a. 48, 36 b. 54, 72 c. 50, 47 d. 47, 50?
In 1991, the population of the People's Republic of China was 1.151 billion, with a growth rate of 1.5% annually.a. Write a recursive formula that models this growth. Let u0 represent the population in 1991.b. Complete a table recording the population for the years 1991 to 2000. c. Define the
Jack planted a mysterious bean just outside his kitchen window. It immediately sprouted 2.56 cm above the ground. Jack kept a careful log of the plant's growth. He measured the height of the plant each day at 8:00 A.M. and recorded these data.a. Define variables and write an exponential equation
For 7a-d, graph the equations on your calculator. a. y = 1.5x b. y = 2x c. y = 3x d. y = 4x e. How do the graphs compare? What points (if any) do they have in common? f. Predict what the graph of y = 6x will look like. Verify your prediction by using your calculator?
Each of the red curves is a transformation of the graph of y = 2x, shown in black. Write an equation for each red curve.a.b. c. d.
For 9a-d, graph the equations on your calculator. a. y = 0.2x b. y = 0.3x c. y = 0.5x d. y = 0.8x e. How do the graphs compare? What points (if any) do they have in common? f. Predict what the graph of y = 0.1x will look like. Verify your prediction by using your calculator?
Rewrite each expression as a fraction without exponents. Verify that your answer is equivalent to the original expression by evaluating each on your calculator. a. 5- 3 b. - 62 c. - 3- 4 d. (- 12) - 2 e. (3/4)2 f. (2/7)-1?
Each of the red curves is a transformation of the graph of y = x3, shown in black. Write an equation for each red curve.a.b. c. d.
Consider the exponential equation y = 47(0.9)x. Several points satisfying the equation are shown in the calculator table. Notice that when x = 0, y = 47. a. The expression 47(0.9)x could be rewritten as 47(0.9)(0.9)x - 1. Explain why this is true. Rewrite 47(0.9)(0.9)x - 1 in the form a · bx -
A ball rebounds to a height of 30.0 cm on the third bounce and to a height of 5.2 cm on the sixth bounce. a. Write two different yet equivalent equations in point-ratio form, y = y1 (bx-x1 using r for the ratio. Let x represent the bounce number, and let y represent the rebound height in
Solve. a. (x-3)3 = 64 b. 256x = 1 / 16 c. (x + 5)3 / (x + 5) = x2 + 25?
A radioactive sample was created in 1980. In 2002, a technician measures the radioactivity at 42.0 rads. One year later, the radioactivity is 39.8 rads. a. Find the ratio of radioactivity between 2002 and 2003. Approximate your answer to four decimal places. b. Let x represent the year, and let y
Name the x-value that makes each equation true. a. 37000000 = 3.7 · 10x b. 0.000801 = 8.01 · 10x b. 47500 = 4.75 · 10x d. 0.0461 = x · 10-2?
Solve this equation for y. Then carefully graph it on your paper. y + 3 / 2 = (x + 4)2
Paul collects these time-distance data for a remote-controlled car.a. Define variables and make a scatter plot of these data. b. Use the median-median line to estimate the car's speed. (Don't do more work than necessary.)
Rewrite each expression in the form an? a. a8 ( a-3 b. b6/b2 c. (c4)5 d. d0 / e-3
State whether each equation is true or false. If it is false, explain why. a. 35 · 42 = 127 b. 100(1.06)x = 106x c. 4x / 4 = 1x d. 6.6 ( 1012 / 8.8 ( 10 - 4 = 7.5 ( 1015
Solve. a. 3x = 1/6 b. (5/3)x = 27/125 c. (1/3)x = 243 d. 5 ( 3x = 5?
Solve each equation. If answers are not exact, approximate to two decimal places. a. x7 = 4000 b. x0.5 = 28 c. x- 3 = 247 d. 5x1/4 + 6 = 10.2 e. 3x- 2 = 2x4 f. -3x1/2 + (4x)1/2 = -1?
Rewrite each expression in the form axn. a. x6 · x6 b. 4x6 · 2x6 c. (-5x3) · (-2x4) d. 72x7/6x2 e. (6x5/3x)3 f. (20x7/4x)-2?
You've seen that the power of a product property allows you to rewrite (a · b)n as an · bn. Is there a power of a sum property that allows you to rewrite (a + b)n as an + bn? Write some numerical expressions in the form (a + b)n and evaluate them. Are your answers equivalent to an + bn always,
Consider this sequence: 72, 72. 25, 72. 5, 72. 75, 73 a. Use your calculator to evaluate each term in the sequence. If answers are not exact, approximate to four decimal places. b. Find the differences between the consecutive terms of the sequence. What do these differences tell you? c. Find the
For 9a-d, graph the equations on your calculator.a. y = x2b. y = x3c. y = x4d. y = x5e. How do the graphs compare? How do they contrast? What points (if any) do they have in common?f. Predict what the graph of y = x6 will look like. Verify your prediction by using your calculator.g. Predict what
Match all expressions that are equivalent. a. 5(x2 b. x2.5 c. 3(3 d. x5/2 f. (1/x)-3 g. ((x)5 h. x3 i. x1/3 j. x2/5
Each of the red curves is a transformation of the graph of the power function y =x3/ 4, shown in black. Write an equation for each red curve.a.b. c. d.
Solve. Approximate answers to the nearest hundredth. a. 9 5(x + 4 = 17 b. (5x4 = 30 c. 43(x2 = (35?
German astronomer Johannes Kepler (1571-1630) discovered in 1619 that the mean orbital radius of a planet, measured in astronomical units (AU), is equal to the time of one complete orbit around the sun, measured in years, raised to a power of 2/3.a. Venus has an orbital time of 0.615. What is its
Discovered by Irish chemist Robert Boyle (1627-1691) in 1662, Boyle's law gives the relationship between pressure and volume of gas if temperature and amount remain constant. If the volume in liters, V, of a container is increased, the pressure in millimeters of mercury (mm Hg), P, decreases. If
Use properties of exponents to find an equivalent expression in the form axn. a. (3x3)x3 b. (2x3)(2x2)3 c. 6x4 / 30x5 d. (4 x2)(3x2)3 e. -72x5y5 / -4x3y (Find an equivalent expression in the form axnym.)
For graphs a-h, write the equation of each graph as a transformation of y = x2 or y = (x?
The town of Hamlin has a growing rat population. Eight summers ago, there were 20 rat sightings, and the numbers have been increasing by about 20% each year. a. Give a recursive formula that models the increasing rat population. Use the number of rats in the first year as u1. b. About how many rat
Identify each function as a power function, an exponential function, or neither of these. (It may be translated, stretched, or reflected.) Give a brief reason for your choice. a. f (x) = 17x5 b. f(t) = t3 + 5 c. g(v) = 200(1.03)v d. h(x) = 2x - 7 e. f(y) = 3 (y - 2 f. f(t) = t2 + 4t + 3 g. h(t) =
Rewrite each expression in the form bn in which n is a rational exponent. a. 6(a b. 10(b8 c. 1/(c d. (5(d)7
Solve each equation and show or explain your step(s). a. 6(a = 4.2 b. 10(b8 = 14.3 c. 1/(c = 0.55 d. (5(d)7 = 23?
Dan placed three colored gels over the main spotlight in the theater so that the intensity of the light on stage was 900 watts per square centimeter (W/cm2). After he added two more gels, making a total of five over the spotlight, the intensity on stage dropped to 600 W/cm2. What will be the
For 6a-d, graph the equations on your calculator. a. y = x1/2 b. y = x1/3 c. y = x1/4 d. y = x1/5 e. How do the graphs compare? What points (if any) do they have in common? f. Predict what the graph of y = x1/7 will look like. Verify your prediction by using your calculator. g. What is the domain
For 7a-d, graph the equations on your calculator. a. y = x1/4 b. y = x2/4 c. y = x3/4 d. y = x4/4 e. How do the graphs compare? What points (if any) do they have in common? f. Predict what the graph of y = x+ will look like. Verify your prediction by using your calculator?
Compare your observations of the power functions in Exercises 6 and 7 to your previous work with exponential functions and power functions with positive integer exponents. How do the shapes of the curves compare? How do they contrast?
Identify each graph as an exponential function, a power function, or neither of these.a.b. c. d.
Solve. a.x5 = 50 b. 3(x = 3.1 c. x2 = - 121?
According to the consumer price index in July 2002, the average cost of a gallon of whole milk was $2.74. If the July 2002 rate of inflation continued, it would cost $3.41 in the year 2024. What was the rate of inflation in July 2002?
A sample of radioactive material has been decaying for 5 years. Three years ago, there were 6.0 g of material left. Now 5.2 g are left. a. What is the rate of decay? b. How much radioactive material was initially in the sample? c. Find an equation to model the decay. d. How much radioactive
In his geography class, Juan makes a conjecture that more people live in cities that are warm (above 50°F) in the winter than live in cities that are cold (below 32°F). In order to test his conjecture, he collects the mean temperatures for January of the 25 largest U.S. cities. These cities
You have solved many systems of two equations with two variables. Use the same techniques to solve this system of three equations with three variables.
Solve. a. x1/ 4 - 2 = 3 b. 4x7 - 6 = - 2 c. 3(x2/3 + 5) = 207? d. 1450 = 800 (1 + x/12)7.8 e. 14.2 = 222.1 ( x3.5
Rewrite each expression in the form axn?a. (27x6)2/3b. (16x8)3/ 4c. (36x- 12)3/2?
A sheet of translucent glass 1 mm thick is designed to reduce the intensity of light. If six sheets are placed together, then the outgoing light intensity is 50% of the incoming light intensity. What is the reduction rate of one sheet in this exponential relation?
Natalie performs a decay simulation using small colored candies with a letter printed on one side. She starts with 200 candies and pours them onto a plate. She removes all the candies with the letter facing up, counts the remaining candies, and then repeats the experiment using the remaining
There is a power relationship between the radius of an orbit, x, and the time of one orbit, y, for the moons of Saturn. (The table at right lists 11 of Saturn's 30 moons.)a. Make a scatter plot of these data.b. Experiment with different values of a and b in the power equation y = axb to find a good
The relationship between the weight in tons, W, and the length in feet, L, of a sperm whale is given by the formula W = 0.000137L3.18. a. An average sperm whale is 62 ft long. What is its weight? b. How long would a sperm whale be if it weighed 75 tons?
In order to estimate the height of an Ailanthus altissima tree, botanists have developed he formula h = 5/3d0.8 where h is the height in meters and d is the diameter in centimeters. a. If the height of an Ailanthus altissima tree is 18 m, find the diameter. b. If the circumference of an Ailanthus
Fat reserves in birds are related to body mass by the formula F = 0.033 · M1.5, where F represents the mass in grams of the fat reserves and M represents the total body mass in grams. a. How many grams of fat reserves would you expect in a 15 g warbler? b. What percent of this warbler's body mass
The data in the table describe the relationship between altitude and air temperature.a. Write a best-fit equation for f (x) that describes the relationship (altitude in meters, temperature in °C). Use at least three decimal places in your answer. b. Use your results from 10a to write the
On Celsius's original scale, freezing corresponded to 100° and boiling corresponded to 0°. a. Write a formula that converts a temperature given by today's Celsius scale into the scale that Celsius invented. b. Explain how you would convert a temperature given in degrees Fahrenheit into a
Here is a paper your friend turned in for a recent quiz in her mathematics class: If it is a four-point quiz, what is your friend's score? For each incorrect answer, provide the correct answer and explain it so that next time your friend will get it right!
In looking over his water utility bills for the past year, Mr. Aviles saw that he was charged a basic monthly fee of $7.18, and $3.98 per thousand gallons (gal) used. a. Write the monthly cost function in terms of the number of thousands of gallons used. b. What is his monthly bill if he uses 8000
Find an exponential function that contains the points (2, 12.6) and (5, 42.525)?
Solve by rewriting with the same base. a. 4x = 83 b. 34x +1 = 9x c. 2x-3 = (1/4)x
Give the equations of two different parabolas with vertex (3, 2) passing through the point (4, 5)?
Solve this system of equations.
Given g(t) = 5 + 2t, find each value. a. g (2) b. g- 1 (9) c. g- 1 (20)?
Match each function with its inverse. a. y = 6 - 2x b. y = 2 - 6/x c. y = - 6(x - 2) d. y = - 6 / x - 2 e. y = - 1 / 2 (x - 6) f. y = 2 / x - 6 g. y = 2 - 1 / 6x h. y = 6 + 2/x?
Given the functions f (x) = - 4 + 0.5(x - 3)2 and g(x) = 3 +(2 (x + 4): a. Find f(7) and g(4). b. What does this imply? c. Find f(1) and g(-2). d. What does this imply? e. Over what domain are f and g inverse functions?
Given f (x) = 4 + (x - 2)3/5: a. Solve for x when f (x) = 12. b. Find f - 1(x) symbolically. c. How are solving for x and finding an inverse alike? How are they different?
Consider the function f (x) = 4 + (x - 2)3/5 given in Exercise 6. a. Graph y = f (x) and use your calculator to draw its inverse. b. Graph the inverse function you found in Exercise 6b. How does it compare to the inverse drawn by your calculator? c. How can you determine whether your answer to
Write each function using f (x) notation, then find its inverse. If the inverse is a function, write it using f - 1(x) notation. a. y = 2x - 3 b. 3x + 2y = 4 c. x2 + 2y = 3?
For each function in 9a and b, find the value of the expressions in i to iv. a. f (x) = 6.34x - 140 b. f (x) = 1.8x + 32 i. f-1(x) ii. f(f-1(15.75)) iii. f-1(f (15.75)) iv. f(f-1(x)) and f-1( f (x)) The equation in 9b will convert temperatures in °C to temperatures in °F. You will use either
1. Rewrite each logarithmic equation in exponential form using the definition of a logarithm. a. log 1000 = x b. log5 625 = x c. log (7 = x d. log8 2 = x e. log5 1/25 = x f. log6 1 = x?
Solve. a. (x - 2)2/ 3 = 49 b. 3x2. 4 - 5 = 16?

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