Find (f + g )(x), (f g)(x), (x),and (x) for each f (x) and g(x). Indicate any restrictions in
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1 Find (f + g )(x), (f g)(x), (x),and (x) for each f (x) and g(x). Indicate any restrictions in domain or range. 2. esolutions Manual - Powered by Cognero Page 1
2 For each pair of functions, find and, if they exist. State the domain and range for each composed function. 4. f = {( 5, 4), (14, 8), (12, 1), (0, 3)} g = {( 2, 4), ( 3, 2), ( 1, 4), (5, 6)} The range of is not a subset of the domain of. So, is undefined. Therefore: Find and, if they exist. State the domain and range for each composed function. 6. esolutions Manual - Powered by Cognero Page 2
3 Find (f + g )(x), (f g)(x), (x),and (x) for each f (x) and g(x). Indicate any restrictions in domain or range. 8. esolutions Manual - Powered by Cognero Page 3
4 esolutions Manual - Powered by Cognero Page 4
5 POPULATION In a particular county, the population of the two largest cities can be modeled by f (x) = 200x + 25 and g(x) = 175x - 15, where x is the number of years since 2000 and the population is in thousands. a. What is the population of the two cities combined after any number of years? b. What is the difference in the populations of the two cities? a. The population of the cities after x years is the sum of the individual populations. b. The difference in the populations of the cities is given by: esolutions Manual - Powered by Cognero Page 5
6 For each pair of functions, find and, if they exist. State the domain and range for each composed function. 18. f = {( 7, 0), (4, 5), (8, 12), ( 3, 6)} g = {(6, 8), ( 12, 5), (0, 5), (5, 1)} esolutions Manual - Powered by Cognero Page 6
7 20. f = {( 4, 14), (0, 6), ( 6, 18), (2, 2)} g = {( 6, 1), ( 18, 13), ( 14, 9), ( 2, 3)} The range of is not a subset of the domain of. So, is undefined. 22. f = {( 1, 11), (2, 2), (5, 7), (4, 4)} g = {(5, 4), (4, 3), ( 1, 2), (2, 3)} The range of is not a subset of the domain of. So, is undefined. esolutions Manual - Powered by Cognero Page 7
8 24. f = {(1, 1), (2, 2), (3, 3), (4, 4)} g = {(1, 4), (2, 3), (3, 2), (4, 1)} The range of is not a subset of the domain of. So, is undefined. The range of is not a subset of the domain of. So, is undefined. 26. f = {(12, 3), (9, 2), (8, 1), (6, 3)} g = {( 1, 5), ( 2, 6), ( 3, 1), ( 4, 8)} esolutions Manual - Powered by Cognero Page 8
9 Find and, if they exist. State the domain and range for each composed function. 28., D = {all real numbers}, R = {all real numbers}, D = {all real numbers}, R = {all real numbers} 30., D = {all real numbers}, R = {y y 14}, D = {all real numbers}, R = {y y 10} esolutions Manual - Powered by Cognero Page 9
10 32., D = {all real numbers}, R = {y y 0.875}, D = {all real numbers}, R = {y y 6.5} 34., D = {all real numbers}, R = {y y 1}, D = {all real numbers}, R = {y y 0} esolutions Manual - Powered by Cognero Page 10
11 36. FINANCE A ceramics store manufactures and sells coffee mugs. The revenue r(x) from the sale of x coffee mugs is given by r(x) = 6.5x. Suppose the function for the cost of manufacturing x coffee mugs is c(x) = 0.75x a. Write the profit function. b. Find the profit on 500, 1000, and 5000 coffee mugs. a. The profit function P(x) is given by where r(x) is the revenue function and c(x) is the cost function. So: b. 38. Perform each operation if f (x) = x 2 + x 12 and g(x) = x 3. State the domain of the resulting function. D = {all real numbers} esolutions Manual - Powered by Cognero Page 11
12 40. If f (x) = 5x, g(x) = 2x + 1, and h(x) = x 2 + 6x + 8, find each value. 42. Substitute x = 3. esolutions Manual - Powered by Cognero Page 12
13 44. Substitute x = Substitute x = 9. esolutions Manual - Powered by Cognero Page 13
14 MULTIPLE REPRESENTATIONS Let f (x) = x 2 and g(x) = x. a. TABULAR Make a table showing values for f (x), g(x), ( f + g) (x), and ( f g)(x). b. GRAPHICAL Graph f (x), g(x), and ( f + g)(x) on the same coordinate grid. c. GRAPHICAL Graph f (x), g(x), and ( f g)(x) on the same coordinate grid. d. VERBAL Describe the relationship among the graphs of f (x), g(x), ( f + g)(x), and ( f g)(x). a. b. esolutions Manual - Powered by Cognero Page 14
15 c. d. Sample answer: For each value of x, the vertical distance between the graph of g(x) and the x-axis is the same as the vertical distance between the graphs of f (x) and (f + g)(x) and between f (x) and (f g)(x). 52. If f (x) = x + 2, g(x) = 4x + 3, and h(x) = x 2 2x + 1, find each value. Substitute x = 3. esolutions Manual - Powered by Cognero Page 15
16 MULTIPLE REPRESENTATIONS You will explore,,, and if f (x) = x and g(x) = x 3. a. Tabular Make a table showing values for,,, and. b. Graphical Use a graphing calculator to graph,,, and on the same coordinate plane. c. Verbal Explain the relationship between and. d. Graphical Use a graphing calculator to graph [f g](x), and [g f ](x) on the same coordinate plane. e. Verbal Explain the relationship between [f g](x), and [g f ](x). a. esolutions Manual - Powered by Cognero Page 16
17 a. b. c. Sample answer: When x is 2 or 4, the functions are equal. d. esolutions Manual - Powered by Cognero Page 17
18 d. e. Sample answer: The functions are translations of the graph of y = x CCSS CRITIQUE Chris and Tobias are finding the composition, where f (x) = x 2 + 2x 8 and g(x) = x Is either of them correct? Explain your reasoning. Tobias is correct. Chris did not substitute g(x) for every x in f (x). esolutions Manual - Powered by Cognero Page 18
19 62. REASONING State whether each statement is sometimes, always, or never true. Explain your reasoning. a. The domain of two functions f (x) and g(x) that are composed g[f (x)] is restricted by the domain of f (x). b. The domain of two functions f (x) and g(x) that are composed g[f (x)] is restricted by the domain of g(x). a. Always; since the range is dependent on the domain, the domain of g[f (x)] is restricted by the domain of f (x). b. Sometimes; when f (x) = 4x and g(x) =, g[f (x)] =. The domain of g(x) restricts the domain of g[f (x)]. When f (x) = 4x 2 and g(x) =, g[f (x)] =. In this case, the domain of g(x) does not restrict the domain of g[f (x)]. 64. What is the value of x in the equation 7(x 4) = 44 11x? A 1 B 2 C 3 D 4 The correct choice is D. esolutions Manual - Powered by Cognero Page 19
20 66. GRIDDED RESPONSE In his first three years of coaching basketball at North High School, Coach Lucas team won 8 games the first year, 17 games the second year, and 6 games the third year. How many games does the team need to win in the fourth year so the coach s average will be 10 wins per year? Let x be the number of games to be won in the fourth year. So: Therefore, the team should win 9 games in the fourth year so that the coach s average will be 10 wins per year. Find all rational zeros of each function. 68. If is a rational zero, then p is a factor of 12 and q is a factor of 2. The possible values of p : The possible values of q: The possible values of : The depressed polynomial is. Solve the quadratic equation. The zeros of the function are esolutions Manual - Powered by Cognero Page 20
21 70. If is a rational zero, then p is a factor of 24 and q is a factor of 1. The possible values of p : The possible values of q: The possible values of : The depressed polynomial is. Solve Therefore, the roots are x = 1, x = 2, x = 4, and x = 3. State the possible number of positive real zeros, negative real zeros, and imaginary zeros of each function. 72. Use Descartes Rule. There are three sign changes in f (x). Therefore, the number of possible positive real zeros is 3 or 1. There is only one sign change in f ( x). Therefore, the number of possible negative real zero is 1. Therefore, the number of possible imaginary zero is 2 or 0. esolutions Manual - Powered by Cognero Page 21
22 74. There are three sign changes in f (x). Therefore, the number of possible positive real zeros is 3 or 1. There is only one sign change in f ( x). Therefore, the number of possible negative real zero is 1. Therefore, the number of possible imaginary zero is 2 or MANUFACTURING A box measures 12 inches by 16 inches by 18 inches. The manufacturer will increase each dimension of the box by the same number of inches and have a new volume of 5985 cubic inches. How much should be added to each dimension? Let x inches be the increase in each dimension of the box. The new volume of the box is given by: Substitute 5965 for V 2. Therefore, 3 inches should be added to each dimension. esolutions Manual - Powered by Cognero Page 22
23 78. Solve each system of equations. The solution is (4, 2, 1). esolutions Manual - Powered by Cognero Page 23
24 80. INTERNET A webmaster estimates that the time, in seconds, to connect to the server when n people are connecting is given by t(n) = 0.005n Estimate the time to connect when 50 people are connecting. Replace n with 50. It takes 0.55 second to connect 50 people. Solve each equation or formula for the specified variable x 2 6xy + 1 = 4, for y 84. D = mv, for m esolutions Manual - Powered by Cognero Page 24
25 86. (x + 2) 2 (y + 5) 2 = 4, for y Use the Square Root Property. esolutions Manual - Powered by Cognero Page 25
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