SOLUTION: The domain of a square root function only includes values for which the radicand is nonnegative.
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1 2. Identify the domain and range of each function. 4. Graph each function. State the domain and range. Find f (0) to determine the lower limit of the range. Therefore: esolutions Manual - Powered by Cognero Page 1
2 6. So: Find f ( 4) to determine the lower limit of the range. esolutions Manual - Powered by Cognero Page 2
3 8. OCEAN The speed that a tsunami, or tidal wave, can travel is modeled by the equation, where v is the speed in kilometers per hour and d is the average depth of the water in kilometers. A tsunami is found to be traveling at 145 kilometers per hour. What is the average depth of the water? Round to the nearest hundredth of a kilometer. Substitute v = 145 and find d. The average depth of the water is about 0.17 km. 10. Graph each inequality. Graph the inequality. esolutions Manual - Powered by Cognero Page 3
4 Identify the domain and range of each function. esolutions Manual - Powered by Cognero Page 4
5 16. Find f ( 2) to determine the lower limit of the range. 18. Find f (6) to determine the upper limit of the range. esolutions Manual - Powered by Cognero Page 5
6 20. Graph each function. State the domain and range. 22. esolutions Manual - Powered by Cognero Page 6
7 24. Find f (4) to determine the lower limit of the range. f(4) = Find f ( 12) to determine the lower limit of the range. f(4) = 3 esolutions Manual - Powered by Cognero Page 7
8 28. Find f (1) to determine the upper limit of the range. f(4) = CCSS MODELING The velocity of a roller coaster as it moves down a hill is, where v is the initial velocity in feet per second and h is the vertical drop in feet. The designer wants the coaster to have a velocity of 90 feet per second when it reaches the bottom of the hill. a. If the initial velocity of the coaster at the top of the hill is 10 feet per second, write an equation that models the situation. b. How high should the designer make the hill? a. Substitute v = 10 and V = 90. b. Solve the equation for h. The designer should make the hill at a height of 125 feet. esolutions Manual - Powered by Cognero Page 8
9 Graph each inequality. 32. Graph the boundary. Since the inequality symbol is >, the boundary should be dashed. The domain is the domain.. Because y is greater than, the shaded region should be above the boundary and within 34. Graph the boundary. Since the inequality symbol is, the boundary line should be solid. The domain is domain.. Because y is less than, the shaded region should be below the boundary and within the esolutions Manual - Powered by Cognero Page 9
10 36. Graph the boundary. Since the inequality symbol is, the boundary line should be solid. The domain is range.. Because y is greater than, the shaded region should be above the boundary and within the 38. Graph the boundary. Since the inequality symbol is <, the boundary line should be dashed. The domain is range.. Because y is less than, the shaded region should be below the boundary and within the esolutions Manual - Powered by Cognero Page 10
11 40. CCSS REASONING After an accident, police can determine how fast a car was traveling before the driver put on his or her brakes by using the equation. In this equation, v represents the speed in miles per hour, f represents the coefficient of friction, and d represents the length of the skid marks in feet. The coefficient of friction varies depending on road conditions. Assume that f = 0.6. a. Find the speed of a car that skids 25 feet. b. If your car is going 35 miles per hour, how many feet would it take you to stop? c. If the speed of a car is doubled, will the skid be twice as long? Explain. a. Substitute f = 0.6 and d = 25. Find v. b. Substitute v = 35 and f = c. No; it is not a linear function. The skid will be 4 times as long. Write the square root function represented by each graph. The graph represents the function. 44. MULTIPLE REPRESENTATIONS In this problem, you will use the following functions to investigate transformations of square root functions. esolutions Manual - Powered by Cognero Page 11
12 a. GRAPHICAL Graph each function on the same set of axes. b. ANALYTICAL Identify the transformation on the graph of the parent function. What values caused each transformation? c. ANALYTICAL Which functions appear to be stretched or compressed vertically? Explain your reasoning. d. VERBAL The two functions that are stretched appear to be stretched by the same magnitude. How is this possible? e. TABULAR Make a table of the rate of change for all three functions between 8 and 12 as compared to 12 and 16. What generalization about rate of change in square root functions can be made as a result of your findings? a. b. f (x): 6 units to the right, 3 units up; to the left, 6 units down; h(x): 3 units to the left, 2 units up. c. Sample answer: f (x) and g(x) appear to be stretched because the graph increases much more quickly than the parent graph. d. Sample answer: They are stretched by the same magnitude because. esolutions Manual - Powered by Cognero Page 12
13 e. The rate of change decreases as the x-values increase for square root functions of the form, where a > PHYSICS Using the function, Einstein s theory of relativity states that the apparent mass m of a particle depends on its velocity v. An object that is traveling extremely fast, close to the speed of light c, will appear to have more mass compared to its mass at rest, m 0. a. Use a graphing calculator to graph the function for a 10,000-kilogram ship for the domain. Use 300 million meters per second for the speed of light. b. What viewing window did you use to view the graph? c. Determine the apparent mass m of the ship for speeds of 100 million, 200 million, and 299 million meters per second. a. b. {0, 400,000,000} scl: 50,000,000 by {0, 100,000} scl: 20,000 c. (100 million, 10,607)(200 million, 13,416)(299 million, 122,577) esolutions Manual - Powered by Cognero Page 13
14 48. REASONING For what positive values of a are the domain and range of the set of real numbers? All positive odd numbers; the set of even numbers would result in a range of nonnegative values. 50. WRITING IN MATH Explain why there are limitations on the domain and range of square root functions. Sample answer: The domain is limited because square roots of negative numbers are imaginary. The range is limited due to the limitation of the domain. 52. WRITING IN MATH Explain why is not a function. To be a function, for every x-value there must be exactly one y-value. For every x in this equation there are two y- values, one that is negative and one that is positive. Also, the graph of does not pass the vertical line test. 54. The expression, is equivalent to A 8x 2 B 8x 3 C 8x 2 D 8x 3 The correct choice is D. esolutions Manual - Powered by Cognero Page 14
15 56. SHORT RESPONSE What is the product of (d + 6) and (d 3)? Determine whether each pair of functions are inverse functions. Write yes or no. 58. The functions f (x) and g(x) are inverses if and only if. Therefore, f and g are inverses. esolutions Manual - Powered by Cognero Page 15
16 60. The functions f (x) and g(x) are inverses if and only if. Therefore, f and g are inverses. esolutions Manual - Powered by Cognero Page 16
17 62. CABLE TV The number of households in the United States with cable TV after 1985 can be modeled by the function C(t) = 43.2t t + 790, where t represents the number of years since a. Graph this equation for the years 1985 to b. Describe the turning points of the graph and its end behavior. c. What is the domain of the function? Use the graph to estimate the range for the function. d. What trends in households with cable TV does the graph suggest? Is it reasonable to assume that the trend will continue indefinitely? a. b. rel. max. between t = 15 and t = 16, and no rel. min.; c. D = {all real numbers}; d. The number of cable TV systems rose steadily from 1985 to Then the number began to decline. The trend may continue for some years, but the number of cable TV systems cannot decline at this rate indefinitely. The number cannot fall below 0. It is not likely that the number would come close to 0 for the foreseeable future; there is no reason to believe that cable TV systems will not be in use. Determine whether each number is rational or irrational neither terminates nor repeats. Therefore, it is irrational. esolutions Manual - Powered by Cognero Page 17
18 The number can be written in the form, for example, as follows: Therefore, it is a rational number. esolutions Manual - Powered by Cognero Page 18
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