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1 FUNCTIONS AND MODELS 1.1 Four Ways to Represent a Function 1. The functions () =  + √2 −  and () =  + √2 −  give exactly the same output values for every input value, so  and  are equal. 2. () = 2 −   − 1 = ( − 1)  − 1 =  for  − 1 6= 0, so  and  [where () = ] are not equal because (1) is undefined and (1) = 1. 3. (a) The point (1 3) is on the graph of , so (1) = 3. (b) When  = −1,  is about −02, so (−1) ≈ −02. (c) ()=1 is equivalent to  = 1 When  = 1, we have  = 0 and  = 3. (d) A reasonable estimate for  when  = 0 is  = −08. (e) The domain of  consists of all -values on the graph of . For this function, the domain is −2 ≤  ≤ 4, or [−2 4]. The range of  consists of all -values on the graph of . For this function, the range is −1 ≤  ≤ 3, or [−1 3]. (f) As  increases from −2 to 1,  increases from −1 to 3. Thus,  is increasing on the interval [−2 1]. 4. (a) The point (−4 −2) is on the graph of , so (−4) = −2. The point (3 4) is on the graph of , so (3) = 4. (b) We are looking for the values of  for which the -values are equal. The -values for  and  are equal at the points (−2 1) and (2 2), so the desired values of  are −2 and 2. (c) () = −1 is equivalent to  = −1. When  = −1, we have  = −3 and  = 4. (d) As  increases from 0 to 4,  decreases from 3 to −1. Thus,  is decreasing on the interval [0 4]. (e) The domain of  consists of all -values on the graph of . For this function, the domain is −4 ≤  ≤ 4, or [−4 4]. The range of  consists of all -values on the graph of . For this function, the range is −2 ≤  ≤ 3, or [−2 3]. (f ) The domain of  is [−4 3] and the range is [05 4]. 5. From Figure 1 in the text, the lowest point occurs at about ( ) = (12 −85). The highest point occurs at about (17 115). Thus, the range of the vertical ground acceleration is −85 ≤  ≤ 115. Written in interval notation, we get [−85 115]. 6. Example 1: A car is driven at 60 mih for 2 hours. The distance  traveled by the car is a function of the time . The domain of the function is { | 0 ≤  ≤ 2}, where  is measured in hours. The range of the function is { | 0 ≤  ≤ 120}, where  is measured in miles. °c 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. 9 NOT FOR SALE INSTRUCTOR USE ONLY © Cengage Learning. All Rights Reserved.
10 ¤ CHAPTER 1 FUNCTIONS AND MODELS Example 2: At a certain university, the number of students  on campus at any time on a particular day is a function of the time  after midnight. The domain of the function is { | 0 ≤  ≤ 24}, where  is measured in hours. The range of the function is { | 0 ≤  ≤ }, where  is an integer and  is the largest number of students on campus at once. Example 3: A certain employee is paid $800 per hour and works a maximum of 30 hours per week. The number of hours worked is rounded down to the nearest quarter of an hour. This employee’s gross weekly pay  is a function of the number of hours worked . The domain of the function is [0 30] and the range of the function is {0 200 400 23800 24000}. 240 pay 0 0.25 0.50 0.75 29.50 29.75 30 hours 2 4 238 236 7. No, the curve is not the graph of a function because a vertical line intersects the curve more than once. Hence, the curve fails the Vertical Line Test. 8. Yes, the curve is the graph of a function because it passes the Vertical Line Test. The domain is [−2 2] and the range is [−1 2]. 9. Yes, the curve is the graph of a function because it passes the Vertical Line Test. The domain is [−3 2] and the range is [−3 −2) ∪ [−1 3]. 10. No, the curve is not the graph of a function since for  = 0, ±1, and ±2, there are infinitely many points on the curve. 11. (a) When  = 1950,  ≈ 138◦C, so the global average temperature in 1950 was about 138◦C. (b) When  = 142◦C,  ≈ 1990. (c) The global average temperature was smallest in 1910 (the year corresponding to the lowest point on the graph) and largest in 2005 (the year corresponding to the highest point on the graph). (d) When  = 1910,  ≈ 135◦C, and when  = 2005,  ≈ 145◦C. Thus, the range of  is about [135, 145]. 12. (a) The ring width varies from near 0 mm to about 16 mm, so the range of the ring width function is approximately [0 16]. (b) According to the graph, the earth gradually cooled from 1550 to 1700, warmed into the late 1700s, cooled again into the late 1800s, and has been steadily warming since then. In the mid-19th century, there was variation that could have been associated with volcanic eruptions. 13. The water will cool down almost to freezing as the ice melts. Then, when the ice has melted, the water will slowly warm up to room temperature. °c 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. NOT FOR SALE INSTRUCTOR USE ONLY © Cengage Learning. All Rights Reserved.
SECTION 1.1 FOUR WAYS TO REPRESENT A FUNCTION ¤ 11 14. Runner A won the race, reaching the finish line at 100 meters in about 15 seconds, followed by runner B with a time of about 19 seconds, and then by runner C who finished in around 23 seconds. B initially led the race, followed by C, and then A. C then passed B to lead for a while. Then A passed first B, and then passed C to take the lead and finish first. Finally, B passed C to finish in second place. All three runners completed the race. 15. (a) The power consumption at 6 AM is 500 MW which is obtained by reading the value of power  when  = 6 from the graph. At 6 PM we read the value of  when  = 18 obtaining approximately 730 MW (b) The minimum power consumption is determined by finding the time for the lowest point on the graph,  = 4 or 4 AM. The maximum power consumption corresponds to the highest point on the graph, which occurs just before  = 12 or right before noon. These times are reasonable, considering the power consumption schedules of most individuals and businesses. 16. The summer solstice (the longest day of the year) is around June 21, and the winter solstice (the shortest day) is around December 22. (Exchange the dates for the southern hemisphere.) 17. Of course, this graph depends strongly on the geographical location! 18. The value of the car decreases fairly rapidly initially, then somewhat less rapidly. 19. As the price increases, the amount sold decreases. 20. The temperature of the pie would increase rapidly, level off to oven temperature, decrease rapidly, and then level off to room temperature. 21. °c 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. NOT FOR SALE INSTRUCTOR USE ONLY © Cengage Learning. All Rights Reserved.