2-6 Special Functions Graph each function. Identify the domain and range. 13. SOLUTION: . . 15. SOLUTION: . Write the piecewise-defined function shown in each graph. 17. eSolutions Manual - Powered by Cognero SOLUTION: Page 1 2-6 Special Functions . Write the piecewise-defined function shown in each graph. 17. SOLUTION: The left portion of the graph is the line g(x) = –x – 4. There is an open circle at (–3, –1), so the domain for this part of the function is . The center portion of the graph is the line g(x) = x + 1. There are closed dots at (–3, –2) and (1, 2), so the domain for this part is . The right portion of the graph is the constant function g(x) = –6. There is an open circle at (4, –6), so the domain for this part is . Write the piecewise function. 19. SOLUTION: The left portion of the graph is the constant function g(x) = 8. There is a closed dot at (–1, 8), so the domain for this part is . The center portion of the graph is the line g(x) = 2x. There are closed dots at (4, 8) and (6, 12), so the domain for this part is . The right portion of the graph is the line g(x) = 2x – 15. There is a circle at (7, –1), so the domain for this part is . eSolutions Manual - Powered by Cognero Write the piecewise function. Page 2 2-6 Special Functions 19. SOLUTION: The left portion of the graph is the constant function g(x) = 8. There is a closed dot at (–1, 8), so the domain for this part is . The center portion of the graph is the line g(x) = 2x. There are closed dots at (4, 8) and (6, 12), so the domain for this part is . The right portion of the graph is the line g(x) = 2x – 15. There is a circle at (7, –1), so the domain for this part is . Write the piecewise function. Graph each function. Identify the domain and range. 21. SOLUTION: D = {all real numbers} R = {all integers} 23. Manual - Powered by Cognero eSolutions SOLUTION: Page 3 D = {all real numbers} R = {all integers} 2-6 Special Functions 23. SOLUTION: The function is defined for all real values of x, so the domain is all real numbers. D = {all real numbers} The function g(x) is twice of a greatest integer function. So, g(x) takes only even integer values. Therefore, the range is R = {all even integers}. Graph each function. Identify the domain and range. 25. SOLUTION: D = {all real numbers } 27. SOLUTION: eSolutions Manual - Powered by Cognero Page 4 D = {all real numbers } 2-6 Special Functions 27. SOLUTION: D = {all real numbers} 29. SOLUTION: D = {all real numbers} 31. CCSS SENSE-MAKING A car’s speedometer reads 60 miles an hour. a. Write an absolute value function for the difference between the car’s actual speed a and the reading on the speedometer. b. What is an appropriate domain for the function? Explain your reasoning. c. Use the domain to graph the function. SOLUTION: a. The absolute value function is . eSolutions Manual - Powered by Cognero b. Since the speed of the car cannot be negative, the appropriate domain for the function is {a | a ≥ 0}. c. Page 5 D = {all real numbers} 2-6 Special Functions 31. CCSS SENSE-MAKING A car’s speedometer reads 60 miles an hour. a. Write an absolute value function for the difference between the car’s actual speed a and the reading on the speedometer. b. What is an appropriate domain for the function? Explain your reasoning. c. Use the domain to graph the function. SOLUTION: a. The absolute value function is . b. Since the speed of the car cannot be negative, the appropriate domain for the function is {a | a ≥ 0}. c. Use each graph to write the absolute value function. 33. SOLUTION: The graph changes its direction at (0, 0). The slope of the line in the interval is –0.5. The slope of the line in the interval is 0.5 . Therefore, the absolute value function is . eSolutions Manual - Powered by Cognero Graph each function. Identify the domain and range. Page 6 2-6 Special Functions Use each graph to write the absolute value function. 33. SOLUTION: The graph changes its direction at (0, 0). The slope of the line in the interval is –0.5. The slope of the line in the interval is 0.5 . Therefore, the absolute value function is . Graph each function. Identify the domain and range. 35. SOLUTION: D = {all real numbers} 37. eSolutions Manual - Powered by Cognero SOLUTION: Page 7 D = {all real numbers} 2-6 Special Functions 37. SOLUTION: D = {all real numbers} . 41. CHALLENGE Graph SOLUTION: 43. OPEN ENDED Write an absolute value function in which f (5) = –3. SOLUTION: Sample answer: 45. SHORT RESPONSE What expression gives the nth term of the linear pattern defined by the table? SOLUTION: eSolutions Manual - Powered by Cognero Page 8 SOLUTION: SampleFunctions answer: 2-6 Special 45. SHORT RESPONSE What expression gives the nth term of the linear pattern defined by the table? SOLUTION: So, the nth term is 3n + 1. 47. NUMBER THEORY Twelve consecutive integers are arranged in order from least to greatest. If the sum of the first six integers is 381, what is the sum of the last six integers? F 345 G 381 H 387 J 417 SOLUTION: Let x be least number in the consecutive integer. Sum of the first six integers = x + (x + 1) + (x + 2) + (x + 3) + (x + 4) + (x + 5) = 6x + 15 Equate 6x + 15 to 381 and solve for x. Therefore, the last 6 integers are 67, 68, 69, 70, 71 and 72. 67 + 68 + 69 + 70 + 71 + 72 = 417 Therefore, option J is the correct answer. Write an equation in slope-intercept form for the line described. 51. passes through (4, 0), parallel to 3x + 2y = 6 SOLUTION: eSolutions - Powered Cognero The Manual slope of the lineby3x + 2y = 6 is . Page 9 67 + 68 + 69 + 70 + 71 + 72 = 417 Therefore, option J is the correct answer. 2-6 Special Functions Write an equation in slope-intercept form for the line described. 51. passes through (4, 0), parallel to 3x + 2y = 6 SOLUTION: The slope of the line 3x + 2y = 6 is . Therefore, the slope of a line parallel to the line 3x + 2y = 6 is . for m in the slope-intercept form. Substitute Substitute 4 and 0 for x and y and solve for b. Therefore, the equation of the line which passes through the point (4, 0) and is parallel to 3x + 2y = 6 is . 2 2 Find each value if f (x) = –4x + 6, g(x) = –x , and h(x) = –2x – 6x + 9. 53. f (2c) SOLUTION: Substitute 2c for x in the function f (x). eSolutions Manual - Powered by Cognero Page 10
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