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1.1 Functions 1 Exercises 1.1 15. It appears that near x 1, the graph goes vertical. A vertical line drawn at that point would touch the graph in multiple locations. However, if the graph doesn’t actually go vertical near x 1, then it is a function. One drawback of reading a graph is that it is sometimes difficult to tell if the graph goes vertical or not. 1. At any given instant in time, you can have only one weight. For example, at age 19 years, 3 months, 2 days, 11 hours, 2 minutes, 27 seconds you have only one weight. Thus your weight is a function of your age. 3. Temperature is a function of the time of day since at any time of the day when the temperature is measured, the measured temperature will be a single value. 5. The input value 182 has two different output values (32 and 47). Therefore, the number of salmon in a catch is not a function of the number of fish caught. 17. Since any vertical line drawn will touch the graph exactly once, the graph is a function. 19. Since any vertical line drawn will touch the graph exactly once, the graph is a function. 21. y x 2 4; 3 x 3, 10 y 1 7. C 4 39.95 4 159.80 The total cost of four pairs of shoes is $159.80. 9. H 2 16 2 120 2 56 Two seconds after he jumped, the cliff diver is 56 feet above the water. 23. y 2 x 2 1; 2 x 2, 3 y 2 11. As shown in the table, E 4 0.06 . In the fourth quarter since December 1999, shares in the tortilla company earned $0.06 per share. 13. As seen in the graph, P 4 18.72 . On November 8, 2001, the closing stock price of the computer company was approximately $18.72. (One drawback of reading a graph is that it is difficult to be precise.) Copyright © Houghton Mifflin Company. All rights reserved. 2 CHAPTER 1 Functions and Linear Models 25. y 5 x x 2 ; x 1 From the graph, it appears that y 4 at x 1. We calculate the exact value of y algebraically: 2 y 5 1 1 5 1 4 27. y 4 x2 4 ; x 2 x 3 2 x 2 3x 6 From the graph, it appears that y 2 at x 2 . We calculate the exact value of y algebraically: y 4 2 4 2 2 2 4 4 4 3 2 2 3 2 6 8 8 6 6 0 0 But division by zero is not a legal operation. Therefore, the function is not defined when x 2 . Graphically speaking, there is a “hole” in the graph at x 2 . 29. The function f ( p) p 2 2 p 1 has the domain of all real numbers. r2 1 has a r2 1 domain of all real numbers, since no value of r will make the denominator equal to zero. 31. The function h r 3t 3 is 4t 4 undefined when the denominator equals zero. 4t 4 0 33. The function g t 4t 4 t 1 The domain of the function is all real numbers except t 1 . That is, t | t 1 . 35. The function f (a) a 3 a 1 is undefined when the radicand is negative. So a 1 0 a 1 The domain of the function is all real numbers greater than or equal to –1. That is, a | a 1 . 2x 6 is x2 3 undefined when the radicand is negative or the denominator is equal to zero. 2x 6 0 37. The function f ( x) 2 x 6 x 3 The denominator is always positive. The domain of the function is the set of all real numbers greater than or equal to –3. That is, x | x 3 . 39. Since it doesn’t make sense to sell a negative number of bags of candy, n is nonnegative. The domain of the candy profit function is the set of whole numbers. That is, n | n is a whole number . Copyright © Houghton Mifflin Company. All rights reserved. 1.2 Linear Functions 41. Using our current calendaring system, the domain of the function is the set of whole numbers between 1 and the current year. For example, in 2006, the domain of the function is n |1 n 2006 . (It is impossible to calculate the average height of z of a person yet to be born in future years.) x 1 is x2 1 undefined whenever the denominator is equal to zero. x2 1 0 43. The function f ( x ) x 1 x 1 0 x 1, x 1 The domain of the function is all real numbers except x 1 and x 1. 45. Yes. Even though the domain value of x 1 is listed twice in the table, it is linked with the same range value, y 6. Exercises 1.2 3 5. The slope of the line passing through (2, 2) and (5, 2) is 22 m 52 0 3 0 7. y-intercept: 0,10 0 5 x 10 10 5 x x 2 x-intercept: 2,0 9. y-intercept: 0,11 0 2 x 11 11 2 x x 5.5 x-intercept: 5.5,0 11. 3 0 y 4 y 4 y 4 y-intercept: 0, 4 1. The slope of the line passing through (2, 5) and (4, 3) is 53 m 24 2 2 1 3x 0 4 3x 4 x x-intercept: 4 3 43 ,0 3. The slope of the line passing through (1.2, 3.4) and (2.7, 3.1) is 3.4 3.1 m 1.2 2.7 0.3 1.5 0.2 Copyright © Houghton Mifflin Company. All rights reserved. 4 CHAPTER 1 Functions and Linear Models 13. The slope of the line passing through (2, 5) and (4, 3) is 53 m 24 2 2 1 y mx b y 1x b 5 1 2 b plugging in 2,5 b7 The slope-intercept form of the line is y x 7 . The standard form of the line is x y 7 . A point-slope form of the line is y 5 1 x 2 . 17. The slope of the line passing through 2, 2 and (5, 2) is m 22 5 2 0 7 0 Since the slope is equal to zero, this line is a horizontal line. The slopeintercept form of the line is y 0 x 2 and is commonly written as y 2 . The standard form of the line is 0 x y 2 and is also often written as y 2 . A point-slope form of the line is y 2 0 x 2 . 19. y 4 x 2 15. The slope of the line passing through (1.2, 3.4) and (2.7, 3.1) is 3.4 3.1 m 1.2 2.7 0.3 1.5 0.2 y mx b y 0.2 x b 3.4 0.2 1.2 b plugging in 1.2,3.4 3.4 0.24 b b 3.64 The slope-intercept form of the line is y 0.2 x 3.64 . The standard form of the line is typically written with integer coefficients. Therefore, y 0.2 x 3.64 21. y 4 0.5 x 2 0.2 x y 3.64 20 x 100 y 364 5 x 25 y 91 (dividing by 4) is the standard form of the line. A point-slope form of the line is y 3.4 0.2 x 1.2 . Copyright © Houghton Mifflin Company. All rights reserved. 1.2 Linear Functions 23. 2 x 3 y 5 5 31. A table containing exactly two points each with different x-values will always represent a linear function. U.S. Average Personal Year Income (in terms of year 2000 dollars) 1989 18,593 1999 28,525 Source: www.census.gov 25. y 23 x 43 The slope is given by 28525 18593 year 2000 dollars m 1999 1989 years 993.2 year 2000 dolllars per year. Between 1989 and 1999, the U.S. average personal income (in year 2000 dollars) increased by an average of $993.20 per year. 33. 27. 0,3 and 3,5 are two points on the line. The slope of the line is 53 m 3 0 2 3 The slope-intercept form of the line 2 is y x 3 . 3 29. This is a vertical line with x-intercept 3,0 . The equation of the line is x 3. Months Take Home Pay (since (dollars) Sep 01) 0 3167.30 1 4350.31 Source: Employee pay stubs The slope is given by 4350.31 3167.30 dollars m 1 0 months 1183.01 dollars per month. Between September 2001 and October 2001, the employee’s take home pay increased at a rate of $1183.01 per month. Copyright © Houghton Mifflin Company. All rights reserved. 6 CHAPTER 1 Functions and Linear Models 35. If the table of data represents a linear function then a linear function passing through two of the points will also pass through all other points in the table. Cost to Dispose of Clean Wood at Enumclaw Transfer Station 500 $18.75 700 $26.25 900 $33.75 1000 $37.50 Source: www.dnr.metrokc.gov Clean Wood (Pounds) The slope of the line passing through 500,18.75 and 700, 26.25 is 26.25 18.75 dollars 700 500 pound 7.5 200 $0.0375 per pound m The equation of the line passing through these points is given by y 0.0375 x b 18.75 0.0375 500 b 18.75 18.75 b b0 y 0.0375 x. We evaluate the linear equation at x 900 and x 1000 . y 0.0375 900 33.75 y 0.0375 1000 37.50 These results match the table data. The data table does represent a linear function. It costs an average of $0.0375 per pound to dispose of clean wood. 37. Let x be the number of servings of WheatiesTM and y be the grams of fiber consumed. We have y 2.1x 3.3 since each serving of cereal contains 2.1 grams of fiber and the banana contains 3.3 grams of fiber. We must solve 8 2.1x 3.3 4.7 2.1x x 2.238 3 In order to consume 8-grams of fiber, you would need to eat 3 servings ( 2 14 cups) of Wheaties along with the large banana. 39. The slope of the line is 7 1 m 44 8 0 undefined. Therefore, the line is a vertical line. Although the y-values change, every point on a vertical line has the same x-value . The x-value of each of the points is x 4 . The equation of the vertical line is x 4 . 41. The slope-intercept form is y mx b and the point-slope form is y y1 m x x1 . For a vertical line, the slope m is undefined and thus may not be substituted into either of the two forms. Copyright © Houghton Mifflin Company. All rights reserved. 1.2 Linear Functions 7 43. Vertical lines are not functions since they fail the Vertical Line Test. However, all non-vertical lines are functions. The slope of the line is y y1 m 2 x2 x1 c 0 b c 0 a c b c a c a b c a . b 45. The slope of the line y 4 is 0 since y 4 may be written as y 0 x 4 . The slope of a line perpendicular to this line has slope 1 m 0 undefined. Therefore, the perpendicular line is a vertical line. Since all points on a vertical line have the same x-values, the equation of the line is x 3 . 47. We can find the x-intercept by plugging in y 0 and the y-intercept by plugging in x 0 . ax b 0 c ax c x c a c The x-intercept is , 0 . a a 0 by c by c c b c The y-intercept is 0, . b y 49. We will first find the equation of the line passing through 2, 7 and 5,13 . 13 7 52 6 3 2 m y 2x b 7 2 2 b 7 4b b3 So y 2 x 3 . We’ll now plug in the point 10,b and solve for b. b 2 10 3 b 23 In order for the table of data to represent a linear function, b must equal 23. Copyright © Houghton Mifflin Company. All rights reserved. 8 CHAPTER 1 Functions and Linear Models 51. The point of intersection of the lines is a, b . This point is the only point that satisfies both of the linear equations. Exercises 1.3 1. a. The scatter plot shows that the data is near linear. b. Using linear regression on the TI-83 Plus, we determine The equation of the line of best fit is y 6.290 x 110.9 . c. m 6.290 means that the Harbor Capital Appreciation Fund share price is dropping at a rate of $6.29 per month. The y-intercept means that in month 0 of 2000 the fund price was $110.94. Since the months of 2000 begin with 1 not 0, the yintercept does not represent the price at the end of January. It could, however, be interpreted as being the price at the end of December 1999. d. This model is a useful tool to show the trend in the stock price between October and December 2000. Since stock prices tend to be volatile, we are somewhat skeptical of the accuracy of data values outside of that domain. 3. a. The scatter plot shows that the data is near linear. b. Using linear regression on the TI-83 Plus, we determine The equation of the line of best fit is y 1165.9 x 80,887 . c. m 1165.9 means that Washington State public university enrollment is increasing by about 1166 students per year. b 80,887 means that, according to the model, Washington State public university enrollment was 80,887 in 1990. Copyright © Houghton Mifflin Company. All rights reserved. 1.3 Linear Models d. The model fits the data extremely well as shown by the graph of the line of best fit. This model could be used by Washington State legislators and university administrators in budgeting and strategic planning. 9 The y-intercept means that in 1995 (year 0), North Carolina was ranked 40th out of the 50 states. (Only positive whole number rankings make sense.) d. This model is not a highly accurate representation of the data as shown by the graph of the line of best fit so it should be used with caution. 5. a. The scatter plot shows that although the data are not linear, they are nonincreasing. b. Using linear regression on the TI-83 Plus, we determine The equation of the line of best fit is y 3.4 x 39.6 . c. m 3.4 means that the per capita income ranking of North Carolina is changing at a rate of 3.4 places per year. That is, the state is moving up in the rankings by about 3 places per year. However, an incumbent government official could use the model in a 2000 reelection campaign as evidence that the state’s economy had improved during his or her tenure in office. The official might also use the model to claim that the trend of improvement will continue if he or she is reelected. 7. a. The minimum fees and the rounding of the weight of the trash make this particular function somewhat complicated. We will first find how much trash may be disposed of for the minimum fees. The amount of trash that can be disposed of for the minimum disposal fee is $13.72 0.1663 ton $82.50 per ton 2000 lbs 0.1663 ton 332.6 lbs 1 ton Copyright © Houghton Mifflin Company. All rights reserved. 10 CHAPTER 1 Functions and Linear Models The largest multiple of twenty that is less than or equal to 332.6 is 320. Therefore, a rounded weight of 320 pounds of trash may be disposed of for the minimum disposal fee. The amount of trash that can be disposed of for the minimum moderate risk waste fee is $1.00 0.3831 ton $2.61 per ton 2000 lbs 0.3831 ton 766.3 lbs 1 ton The largest multiple of twenty that is less than or equal to 766.3 is 760. For each of the following functions, we let x represent the weight of the trash rounded to the nearest 20 pounds. The cost of disposing 320 pounds of trash or less (including tax) is C ( x ) 13.72 11.036 $15.25 The cost of disposing between 340 and 760 pounds of trash (including tax) is x C x 82.50 1 1.036 2000 0.04125 x 11.036 x C x 82.50 2.61 1.036 2000 0.04256 x 1.036 0.04409 x Combining the individual functions into a single piecewise cost function we have 0 x 320 15.25 C x 0.04274 x 1.036 , 340 x 760 0.04409 x 780 x For values of x 780 , C x is directly proportional to x. b. We must first round the weight of the trash to the nearest 20 pounds. One way to do this is to divide the weight of the trash by 20, round the number to the nearest whole number, and multiply the result by 20. That is, 230 11.5 20 12 12 20 240 Since this weight is below 320 pounds, the minimum $15.25 fee will be charged. c. How much will it cost to drop off 513 pounds of trash? 0.04274 x 1.036 The cost of disposing of 780 pounds of trash or more is 513 25.65 20 26 12 26 520 C 520 0.04274 520 1.036 $23.26 Copyright © Houghton Mifflin Company. All rights reserved. 1.3 Linear Models d. The cost per pound is lowest when 780 or more pounds of trash are disposed of. As a construction company, we would try to keep the weight of our trash deliveries at or above 780 pounds. 9. a. Let C be the total cost (in dollars) of using t anytime minutes of phone time during a given month. (If a fraction of a minute is used, t is the next whole number value. For example, if 3.2 anytime minutes are used then t 4 .) The $29.99 fee covers the first 200 minutes. The $0.40 per minute fee is only charged on the minutes used after the first 200 minutes. Thus we have 29.99, 0 t 200 C t 29.99 0.40 t 200 , t 200 which may be rewritten as 29.99, 0 t 200 C t 0.4t 50.01, t 200 b. For the Qwest plan we have C 300 0.35 300 22.51 $82.49 For the Sprint plan we have C 300 0.4 300 50.01 $69.99 11 11. a. Let t be the production year of a Toyota Land Cruiser 4-Wheel DriveTM and V be the value of the vehicle in 2001. We have 1995, 21125 and 2000, 43650 . We must find the linear function passing through these points. We have 43650 21125 m 2000 1995 4505 so V 4505t b . We solve for b using the point 2000, 43650 . 43650 4505 2000 b b 8,966,350. The linear model is V 4505t 8966350 . b. We have V 1992 4505 1992 8966350 7610 According to the model, a 1992 Toyota Land Cruiser was valued at $7610 in 2001. c. The model substantially underestimated the value of a 1992 Land Cruiser. If we draw a scatter plot of the value of a 1992, 1995, and 2000 Land Cruisers in 2001, we see that the vehicle value function is not linear. The Sprint plan is the best deal for a customer who uses 300 anytime minutes. If a plan was selected that offered 300 anytime minutes for a higher basic fee, it would likely cost the customer even less. Copyright © Houghton Mifflin Company. All rights reserved. 12 CHAPTER 1 Functions and Linear Models 13. a. Let F be the number of grams of fat in x chef salads. We have F 8x . The number of fat grams is directly proportional to the number of chef salads. b. We can consume up to 80 grams of fat. 80 8x x 10 Thus a person on a 2500-calorie diet can consume up less than 10 salads. c. One package of ranch salad dressing contains 18 grams of fat. F 8 x 18 80 8 x 18 62 8 x x 7.75 A person on a 2500-calorie diet can consume up to 7.75 salads before exceeding the fat requirement. d. F 8 x 18 65 8 x 18 15. Fees imposed on enplaning passengers include the flight segment tax ($3), the PFC (up to $4.50), and a security fee ($2.50). That is, up to $10 in fees is charged for each segment. Let s be the number of segments and C be the maximum total cost of the ticket (in dollars) from Seattle to Phoenix. We have C 10s 222 . Note that s 1 . 17. A business determines that the revenue generated by selling x cups of coffee, y bagels, and z muffins is given by R ax by cz . a is the price of a cup of coffee, b is the price of a bagel, and c is the price of a muffin. 19. The equation of the line of best fit for the data in the table is y 0 . x y 0 –1 1 1 2 0 3 0 4 1 5 –1 6 0 47 8 x x 5.875 A person on a 2000-calorie diet can consume up to 5.875 salads before exceeding the fat requirement. The correlation coefficient r 0 indicates that the model doesn’t fit the data well. Copyright © Houghton Mifflin Company. All rights reserved. 1.3 Linear Models 13 23.The line of best fit is y 1 . This model fits the table of data perfectly. In addition, we can see from the scatter plot that the model doesn’t fit well. 21. The slope-intercept form of a line passing through the origin is y mx 0 mx This implies that y is directly proportional to x. If the linear model y mx fits the original data well, it is likely that the dependent variable of the original data is directly proportional to the independent variable. However, if the linear model y mx does not fit the data well, the dependent and independent variables of the original data are not directly proportional. For example, consider the table of data x 0 1 2 3 4 5 y 0 1 -1 -1 1 0 The line of best fit is y 0 x which indicates that x and y are directly proportional. However, we can see from the table that y is not directly proportional to x. However, from the calculator display we see that r is undefined. The correlation coefficient is formally defined as n xy x y r 2 2 n x2 x n y 2 y The symbol tells us to add each of the terms. Since we have four data points, n 4 for this data set. We will calculate each term individually and then substitute the result into the correlation coefficient formula. n xy 4 0 1 1 1 2 1 3 1 4 6 24 x y 0 1 2 31 1 1 1 6 4 24 n x 2 4 02 12 22 32 4 14 56 Copyright © Houghton Mifflin Company. All rights reserved. 14 CHAPTER 1 Functions and Linear Models x 0 1 2 3 2 2 25. 36 n y 2 4 12 12 12 12 4 4 16 y 1 1 1 1 2 2 42 16 Thus we have 24 24 r 56 36 16 16 0 0 Because we can’t divide by zero and there is a zero in the denominator of the expression, the correlation coefficient r is undefined. From the scatter plot, it appears that the first four points are in a line and the last three points are in a line. The slope of the first line is m 2 . The slope of the second line is m 0.5 . Both lines pass through 4,2 . The equation of the first line is y 2 x b 2 2 4 b 2 8 b b 10 So y 2 x 10. The equation of the second line is y 0.5 x b 2 0.5 4 b 2 2b b0 So y 0.5 x The piecewise function is 2 x 10 2 x 4 y 4 x 10 0.5x Copyright © Houghton Mifflin Company. All rights reserved. Chapter 1 Review Exercises Review Exercises Section 1.1 1. C 2 49.95 2 99.90 The cost to buy two pairs of shoes is $99.90. 3. H 2 16 2 100 2 64 100 36 The cliff diver is 36 feet above the water 2 seconds after he jumps from a 100-foot cliff. 5. f ( p) p 2 9 p 15 The domain of the function is all real numbers. r 1 r2 1 Since r 1 and r 1 make the denominator equal zero, the domain of the function is all real numbers except r 1 and r 1 . 7. C r 2 9. At t 2 , P 21.03 . The stock price of the computer company at the end of the day two days after Dec 16, 2001, was about $21.03. 11. The graph passes the Vertical Line Test, since no vertical line drawn will cross the graph more than once. Thus, the graph represents a function. 15 Section 1.2 13. (–3, –4) and (0, –2) 2 4 m 0 3 2 3 15. (7, 11) and (8, 0) 0 11 m 87 11 1 11 17. y 3x 18 The y-intercept is 0,18 . 0 3x 18 3x 18 x6 The x-intercept is 6,0 . 19. (2, 5) and (4, 3) 35 42 1 y 1x b m 5 1 2 b b7 So y 1x 7 In standard form, x y 7 . Copyright © Houghton Mifflin Company. All rights reserved. 16 CHAPTER 1 Functions and Linear Models Section 1.3 21. a. Let x be the number of large orders of french fries consumed. The amount of fat consumed is given by F x 29 x grams. b. Let y be the number of Big N’ TastyTM sandwiches consumed. The amount of fat consumed is given by G y 32 y grams. c. The total amount of fat consumed by eating z combination meals is H z 32 29 z . b. We have v t 7312.5t 14,558,975 . v 1999 7312.5 1999 14,558,975 $58,712.50 c. The linear model was extremely effective at accurately predicting the value of the 1999 MercedesBenz Roadster 2-door SL500. The $212.50 difference between the predicted value of the model and the NADA guide average value was negligible. H 1 32 29 1 61 Only one combination meal may be eaten. Eating more than one will exceed the RDA for fat (at most 65 grams). 23. a. Let t be the production year of the Mercedes-Benz Roadster 2-door SL500TM and let v t be the value of the car in 2001. We have v t t 1998 $51,400 2000 $66,025 The slope of the car value function is 66025 51400 m 2000 1998 $7312.5 per year We have v t 7312.5t b 51400 7312.5 1998 b b 14,558,975 So v t 7312.5t 14,558,975 is the linear model for the data. Copyright © Houghton Mifflin Company. All rights reserved.