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D = {–2, 1, 4}, R = {–1, 2, 3, 5}; 2-1 Relations and Functions CCSS STRUCTURE State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither. The relation is not a function because 1 is mapped to both 2 and 5. 3. SOLUTION: The members of the domain are the x-values of the relation while the members of the range are the yvalues. 1. D = {–2, 1, 4, 8}, R = {–4, –2, 6}; SOLUTION: The left side of the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {–2, 5, 6}, R = { –8, 1, 3}; Each element of the domain is paired with exactly one element of the range. So, the relation is a function. Each element of the domain is paired with exactly one element of the range. So, the relation is a function. The function is onto because each element of the range corresponds to an element of the domain. 4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years. The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain. 2. SOLUTION: The members of the domain are the x-values of the relation while the members of the range are the yvalues. a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function? D = {–2, 1, 4}, R = {–1, 2, 3, 5}; SOLUTION: a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2} The relation is not a function because 1 is mapped to both 2 and 5. b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)} eSolutions Manual - Powered by Cognero 3. SOLUTION: c. The domain is a set of individual points. So the relation is discrete. Page 1 d. The relation is a function as each element of the domain is paired with exactly one element of the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), and (26, 24.6), (27, 30.2)} 2-1 Relations Functions c. The domain is a set of individual points. So the relation is discrete. d. The relation is a function as each element of the domain is paired with exactly one element of the range. Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous. 5. SOLUTION: To graph the equation, substitute different values of x in the equation and solve for y. Then connect the points. x 0 1 2 3 -1 -2 -3 y = 5x + 4 4 5 14 19 -1 -6 -11 Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous. 5. SOLUTION: To graph the equation, substitute different values of x in the equation and solve for y. Then connect the points. The members of the domain are the x-values of the relation while the members of the range are the yvalues. D = {all real numbers}; R = {all real numbers}; x 0 1 2 3 -1 -2 -3 y = 5x + 4 4 5 14 19 -1 -6 -11 No vertical line intersects the graph in more than one point. So the graph is a function. The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain. The graph of the function is a line. So the function is continuous. 6. Manual - Powered by Cognero eSolutions The members of the domain are the x-values of the relation while the members of the range are the y- SOLUTION: To graph the equation, substitute different values of x in the equation and solve for y. Then connect thePage 2 points. correspond to an element of the domain. corresponds to an element of the domain. The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous. The graph of the function is a line. So the function is 2-1 Relations continuous.and Functions 6. 7. SOLUTION: To graph the equation, substitute different values of x in the equation and solve for y. Then connect the points. SOLUTION: To graph the equation, substitute different values of x in the equation and solve for y. Then connect the points. x y = -4x - 2 0 1 2 3 -1 -2 -3 x -2 -6 -10 -14 2 6 10 y = 3x 0 1 2 3 -1 -2 -3 2 0 3 12 27 3 12 27 The members of the domain are the x-values of the relation while the members of the range are the yvalues. D = {all real numbers}; R = {all real numbers}; The members of the domain are the x-values of the relation while the members of the range are the yvalues. D = {all real numbers}; R = {all real numbers}; No vertical line intersects the graph in more than one point. So the graph is a function. No vertical line intersects the graph in more than one point. So the graph is a function. The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain. The function is neither one-to-one nor onto because the elements in the domain do not have unique images and the negative numbers are left unmapped. The domain has an infinite number of elements and the relation can be graphed using a smooth curve. So the relation is continuous. The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous. 8. 7. SOLUTION: To graph the equation, substitute different values of x in the equation and solve for y. Then connect the points. SOLUTION: The graph of the equation is a vertical line through (7, 0). x y = 3x 2 eSolutions Manual - Powered by Cognero 0 1 2 0 3 12 Page 3 images and the negative numbers are left unmapped. The domain has an infinite number of elements and 2-1 Relations the relationand canFunctions be graphed using a smooth curve. So the relation is continuous. 8. SOLUTION: The graph of the equation is a vertical line through (7, 0). State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is oneto-one, onto, both, or neither. 11. SOLUTION: The members of the domain are the x-values of the relation while the members of the range are the yvalues. In this equation x is always 7 for any value of y. D = {7}; R = {all real numbers}; The only element in the domain is mapped to all the elements in the range. So it is not a function. D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4} 1.2 is mapped to both –1 and 4. So the relation is not a function. The domain has a finite number (1) of elements, so the relation is not continuous. Evaluate each function. 9. SOLUTION: Replace x by –3. 12. SOLUTION: The members of the domain are the x-values of the relation while the members of the range are the yvalues. D = {–8, 2, 4}; R = {–6, –4, 14}; –8 is mapped to both –4 and 14. So the relation is not a function. 10. 13. {(–3, –4), (–1, 0), (3, 0), (5, 3)} SOLUTION: Replace x with 5. SOLUTION: The members of the domain are the x-values of the relation while the members of the range are the yvalues. D = {–3, –1, 3, 5}; R = {–4, 0, 3} Each element of the domain is paired with exactly one element in the range. So, the relation is a function. State the domain and range of each relation. Then determine whether each relation is a eSolutions Manual by Cognero determine if it is onefunction . If- Powered it is a function, to-one, onto, both, or neither. The function is onto because each element of the range corresponds to an element of the domain. Page 4 14. POLITICS The table below shows the population of a. Scale each axis of the graph by 5. Since population is on the horizontal axis, these are the x-values of the relation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph. D = {–8, 2, 4}; R = {–6, –4, 14}; 2-1 Relations andtoFunctions –8 is mapped both –4 and 14. So the relation is not a function. 13. {(–3, –4), (–1, 0), (3, 0), (5, 3)} SOLUTION: The members of the domain are the x-values of the relation while the members of the range are the yvalues. D = {–3, –1, 3, 5}; R = {–4, 0, 3} Each element of the domain is paired with exactly one element in the range. So, the relation is a function. The function is onto because each element of the range corresponds to an element of the domain. 14. POLITICS The table below shows the population of several states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain your reasoning. b. The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53} c. The domain is a set of individual points. So the relation is discrete. d. The relation is a function because each domain value is paired with only one range value. CCSS STRUCTURE Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous. 15. SOLUTION: a. Scale each axis of the graph by 5. Since population is on the horizontal axis, these are the x-values of the relation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph. SOLUTION: To graph, substitute values for x into the equation and solve for y. Draw a smooth curve through these points. x y = -3x + 2 0 1 2 3 -1 -2 -3 2 -1 -4 -7 5 8 11 eSolutions Manual - Powered by Cognero Page 5 c. The domain is a set of individual points. So the relation is discrete. The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous. d. The relation 2-1 Relations and Functions is a function because each domain value is paired with only one range value. CCSS STRUCTURE Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous. 16. SOLUTION: To graph, substitute values for x into the equation and solve for y. Draw a smooth curve through these points. 15. x SOLUTION: To graph, substitute values for x into the equation and solve for y. Draw a smooth curve through these points. x y = -3x + 2 0 1 2 3 -1 -2 -3 y = 0.5x - 3 0 1 2 3 -1 -2 -3 2 -1 -4 -7 5 8 11 -3 -2.5 -2 -1.5 -3.5 -5 -4.5 The members of the domain are the x-values of the relation while the members of the range are the yvalues. D = {all real numbers}; R = {all real numbers}; The members of the domain are the x-values of the relation while the members of the range are the yvalues. D = {all real numbers}; R = {all real numbers}; No vertical line intersects the graph in more than one point. So the equation is a function. The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain. No vertical line intersects the graph in more than one point. So the equation is a function. The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain. The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous. The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous. 17. SOLUTION: To graph, substitute values for x into the equation and solve for y. Draw a smooth curve through these points. 16. y = 2x 2 x eSolutions Manual - Powered by Cognero SOLUTION: To graph, substitute values for x into the equation and solve for y. Draw a smooth curve through these 0 1 2 0 2 8 Page 6 corresponds to an element of the domain. numbers are left unmapped. The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous. The domain has an infinite number of elements and 2-1 Relations the relationand canFunctions be graphed with a solid straight line. So the relation is continuous. 18. 17. SOLUTION: To graph, substitute values for x into the equation and solve for y. Draw a smooth curve through these points. SOLUTION: To graph, substitute values for x into the equation and solve for y. Draw a smooth curve through these points. x y = 2x 2 0 1 2 3 -1 -2 -3 x 0 2 8 18 2 8 18 y = -5x 2 0 1 2 3 -1 -2 -3 0 -5 -20 -45 -5 -20 -45 The members of the domain are the x-values of the relation while the members of the range are the yvalues. D = {all real numbers}; The members of the domain are the x-values of the relation while the members of the range are the yvalues. D = {all real numbers}; No vertical line intersects the graph in more than one point. So the equation is a function. No vertical line intersects the graph in more than one point. So the equation is a function. The function is not one-to-one because each element of the domain is not paired with a unique element of the range. The function is not one-to-one because each element of the domain is not paired with a unique element of the range. The function is not onto because the negative numbers are left unmapped. The function is not onto because the positive numbers are left unmapped. The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous. The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous. 19. 18. SOLUTION: To graph, substitute values for x into the equation and solve for y. Draw a smooth curve through these points. eSolutions Manual - Powered by Cognero SOLUTION: To graph, substitute values for x into the equation and solve for y. Draw a smooth curve through these points. Page 7 x y = -5x 2 x y = 4x 2 - 8 numbers are left unmapped. than –8 are left unmapped. The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous. The domain has an infinite number of elements and 2-1 Relations the relationand canFunctions be graphed with a smooth curve. So the relation is continuous. 20. 19. SOLUTION: To graph, substitute values for x into the equation and solve for y. Draw a smooth curve through these points. SOLUTION: To graph, substitute values for x into the equation and solve for y. A few of the points on the graph are (0, - 1), (1, -4), (-1, 2), , (2, -25), and (-2, 23). Draw a smooth curve through these points. 2 x y = 4x - 8 0 1 2 3 -1 -2 -3 -8 -4 8 28 -4 8 28 The members of the domain are the x-values of the relation while the members of the range are the yvalues. D = {all real numbers}; R = {all real numbers}; No vertical line intersects the graph in more than one point. So the equation is a function. The members of the domain are the x-values of the relation while the members of the range are the yvalues. D = {all real numbers}; No vertical line intersects the graph in more than one point. So the equation is a function. The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous. The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain. The function is not one-to-one because each element of the domain is not paired with a unique element of the range. Evaluate each function. 21. The function is not onto because the numbers less than –8 are left unmapped. SOLUTION: Replace x with –8. The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous. 20. SOLUTION: To graph, substitute values for x into the equation and solveManual for y.- Powered A few of points on the graph are (0, eSolutions bythe Cognero 1), (1, -4), (-1, 2), , (2, -25), and (-2, 23). Draw a smooth curve through these 22. SOLUTION: Replace x with 2.5. Page 8 Replace x with –8. Replace x with 2.5. 2-1 Relations and Functions 23. DIVING The table below shows the pressure on a diver at various depths. 22. SOLUTION: a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning. Replace x with 2.5. SOLUTION: a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)} 23. DIVING The table below shows the pressure on a diver at various depths. a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning. b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. SOLUTION: a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)} b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D = ; R= . The relation is continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet. d. Each domain value is paired with only one range value. So the relation is a function. c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D = ; R= . The relation is continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet. d. Each domain value is paired with only one range value. So the relation is a function. eSolutions Manual - Powered by Cognero Find each value if and 24. SOLUTION: f (x) = 3x + 2 Page 9 such as 10 feet. . d. Each domain value is paired with only one range value. So the is a function. 2-1 Relations andrelation Functions Find each value if 27. and SOLUTION: 24. Replace x with –6. SOLUTION: f (x) = 3x + 2 Replace x with –5. 28. SOLUTION: 25. Replace x with 3. SOLUTION: f (x) = 3x + 2 Replace x with 9. 29. SOLUTION: 26. Replace x with 8 . SOLUTION: Replace x with –3. . 30. SOLUTION: 27. SOLUTION: Replace x with . Replace x with –6. eSolutions Manual - Powered by Cognero 28. Page 10 31. . 2-1 Relations and Functions 30. 32. SOLUTION: SOLUTION: Replace x with . Replace x with . 31. SOLUTION: Replace x with . 33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides to download 3 more podcasts each month. The function P(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months? SOLUTION: Replace t with 8. 32. After 8 months Chaz will have 39 podcasts. SOLUTION: 34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions. Replace x with . eSolutions Manual - Powered by Cognero a. GRAPHICAL Graph each function on a separate graphing calculator screen. b. TABULAR Use the graphs to create a tablePage 11 showing the number of times a horizontal line could intersect the graph of each function. List all 2-1 Relations and Functions b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities. j(x) = x 3 c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning. b. d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning. e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto. SOLUTION: a. f (x) = x 2 c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not. e. g(x) = 2x 35. CCSS CRITIQUE Omar and Madison are finding f (3d) for the function Is either of them correct? Explain your reasoning. h(x) = x 3 - 3x 2 - 5x + 6 j(x) = x 3 eSolutions Manual - Powered by Cognero SOLUTION: Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct. 36. CHALLENGE Consider the functions f (x) and and g(a) = 33, while f (b) = 31 and g (b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x). Page 12 SOLUTION: Sample answer: Organize the given information into a table. SOLUTION: Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square and the 3Functions before multiplying by –4 so Omar is 2-1 Relations correct. 36. CHALLENGE Consider the functions f (x) and and g(a) = 33, while f (b) = 31 and g (b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x). SOLUTION: Sample answer: Organize the given information into a table. f (x) g(x) f (a) = 19 g(a) = 33 f (b) = 31 g(b) = 51 a = 5, b = 8 Analyze the information given about f (x). f (x) f (a) = f (5) = 4(5) = f (x) = f (a) = 4(5) 19 19 20 4x – 1 – 1 = 20 f (b) = f (8) = 4(8) = f (x) = f (b) = 4(8) 31 31 32 4x – 1 – 1 = 31 If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b). Next, analyze the information given about g(x). g(x) g(a) = 6 g(a) = g(5) = 6(5) = g(x) = (5) + 3 = 33 33 30 6x + 3 33 g(b) = 6 g(b) = g(8) = 6(8) = g(x) = (8) + 3 = 51 51 48 4x + 3 51 If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b). So the functions are: . Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test. 38. OPEN ENDED Graph a relation that can be used to represent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M. SOLUTION: a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground. b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increases as a child, then more steeply, finally leveling off and remaining constant. 37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning. SOLUTION: Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test. 38. OPEN ENDED Graph a relation that can be used to represent each of the following. a. the height of a baseball that is hit into the outfield eSolutions Manual - Powered by Cognero b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets. Page 13 d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature 2-1 Relations and Functions gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets. SOLUTION: Sample answer: A relation is a function if each xvalue only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with more than one y-value, so the relation is not a function. 41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes? A g = 19,500 – 6m 39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one as well. SOLUTION: Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range. 40. WRITING IN MATH Explain why the vertical line test can determine if a relation is a function. SOLUTION: Sample answer: A relation is a function if each xvalue only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with more than one y-value, so the relation is not a function. B g = 19,500 + 6m C D SOLUTION: Number of gallons of water in the pool = 19,500. Patricia drains the water at a rate of 6 gallons per minute. In m minutes, she can drain 6m gallons of water. So the number of gallons of water remaining in the pool after m minutes is given by: g = 19,500 – 6m The correct choice is A. 42. SHORT RESPONSE Look at the pattern below. 41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes? If the pattern continues, what will the next term be? SOLUTION: A g = 19,500 – 6m Each term of the pattern is obtained by adding the previous term. to B g = 19,500 + 6m Next term = C 43. GEOMETRY Which set of dimensions represents a triangle similar to the triangle shown below? D SOLUTION: Number of gallons of water in the pool = 19,500. minute. F 1 unit, 2 units, 3 units G 7 units, 11 units, 12 units H 10 units, 23 units, 24 units J 20 units, 48 units, 52 units SOLUTION: Patricia drains the water at a rate eSolutions Manual - Powered by Cognero of 6 gallons per Page 14 Each term of the pattern is obtained by adding to the previous term. 2-1 Relations Next term and = Functions The correct choice is C. Solve each inequality. 43. GEOMETRY Which set of dimensions represents a triangle similar to the triangle shown below? 45. SOLUTION: F 1 unit, 2 units, 3 units G 7 units, 11 units, 12 units H 10 units, 23 units, 24 units J 20 units, 48 units, 52 units SOLUTION: No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H. 46. SOLUTION: The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4. So the correct choice is J. 44. ACT/SAT If to g(x + 1)? A. 1 which expression is equal 2 B. x + 1 2 C. x + 2x + 1 47. SOLUTION: 2 D. x – x 2 E. x + x + 1 SOLUTION: Replace x by x + 1. This implies: The correct choice is C. Solve each inequality. 45. 48. CLUBS Mr. Willis is starting a chess club at his high school. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequality representing the situation. SOLUTION: eSolutions Manual - Powered by Cognero SOLUTION: Let x represent the number of members. Page 15 Ling can buy a maximum of 8 shirts. 2-1 Relations and Functions 48. CLUBS Mr. Willis is starting a chess club at his high school. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequality representing the situation. Solve each equation. Check your solutions. 50. SOLUTION: This implies: SOLUTION: Let x represent the number of members. 49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buy shirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy. 51. SOLUTION: SOLUTION: Let x be the number of shirts Ling can buy. Each shirt costs $15. So: Ling can buy a maximum of 8 shirts. This implies: 52. SOLUTION: Solve each equation. Check your solutions. 50. SOLUTION: This implies: This implies: Simplify each expression. eSolutions Manual - Powered by Cognero 51. Page 16 53. SOLUTION: 2-1 Relations and Functions Simplify each expression. So the solution is x = 6. 57. SOLUTION: 53. SOLUTION: Substitute a = 4 in the original equation. 54. SOLUTION: So the solution is a = 4. 55. 58. SOLUTION: SOLUTION: Solve each equation. Check your solutions. 56. Substitute x = –2 in the original equation. SOLUTION: So the solution is x = –2. Substitute x = 6 in the original equation. 59. SOLUTION: So the solution is x = 6. eSolutions Manual - Powered by Cognero 57. SOLUTION: Substitute b = –4 in the original equation. Page 17 2-1 Relations and Functions So the solution is x = –2. 59. So the solution is x = 3. 61. SOLUTION: SOLUTION: Substitute b = –4 in the original equation. Substitute y = –4 in the original equation. So the solution is b = –4. 60. So the solution is y = –4. SOLUTION: 62. SOLUTION: Substitute x = 3 in the original equation. Substitute c = 6 in the original equation. So the solution is x = 3. So the solution is c = 6. 63. 61. SOLUTION: SOLUTION: eSolutions Manualy- Powered Substitute = –4 inby theCognero original equation. Substitute d = –6 in the original equation. Page 18 2-1 Relations and Functions So the solution is c = 6. 63. SOLUTION: Substitute d = –6 in the original equation. So the solution is d = –6. 64. SOLUTION: Substitute y = 3 in the equation. So the solution is y = 3. eSolutions Manual - Powered by Cognero Page 19