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14. The factory sales f (in millions of dollars) of digital cameras in the US from 1998 through 2003 are shown in the table below. The time t (in years) is given with t = 8 corresponding to 1998. Year, t 8 Sales, f(t) 519 9 1209 a) Does f¹(t) exist? 10 1825 11 1972 12 2794 13 3421

14. The factory sales f (in millions of dollars) of digital cameras in the US from 1998 through 2003 are shown in the table below. The time t (in years) is given with t = 8 corresponding to 1998. Year, t 8 Sales, f(t) 519 9 1209 a) Does f¹(t) exist? 10 1825 11 1972 12 2794 13 3421

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Sales of Digital Cameras in the US (1998-2003)** The table below shows the factory sales \( f \) (in millions of dollars) of digital cameras in the US from 1998 through 2003. The time \( t \) (in years) is given with \( t = 8 \) corresponding to 1998. | Year, \( t \) | 8 | 9 | 10 | 11 | 12 | 13 | |---------------|-----|------|------|------|------|------| | Sales, \( f(t) \) | 519 | 1209 | 1825 | 1972 | 2794 | 3421 | ### Questions a) Does \( f^{-1}(t) \) exist? b) If \( f^{-1}(t) \) exists, what do the variables represent in the inverse function? c) If \( f^{-1}(t) \) exists, find \( f^{-1}(1825) \). d) If the table was extended to 2004 and if the factory sales of digital cameras for that year was $2794 million, would \( f^{-1}(t) \) exist? ### Explanation **a)** To determine if \( f^{-1}(t) \) exists, the function must be one-to-one. This means each \( f(t) \) should map to a unique year \( t \). **b)** If \( f^{-1}(t) \) exists, the variables would represent: \( f^{-1}(s) = t \), where \( s \) (sales value in millions) is mapped to \( t \) (the corresponding year). **c)** Looking up 1825 in the sales column of the table, the corresponding year \( t \) is 10. Thus, \( f^{-1}(1825) = 10 \). **d)** If a sales value is repeated for a different year, the function would not be one-to-one, and \( f^{-1}(t) \) would not exist. Since 2794 already corresponds to \( t = 12 \), having 2794 for an extended year would imply the inverse does not exist.
Transcribed Image Text:**Sales of Digital Cameras in the US (1998-2003)** The table below shows the factory sales \( f \) (in millions of dollars) of digital cameras in the US from 1998 through 2003. The time \( t \) (in years) is given with \( t = 8 \) corresponding to 1998. | Year, \( t \) | 8 | 9 | 10 | 11 | 12 | 13 | |---------------|-----|------|------|------|------|------| | Sales, \( f(t) \) | 519 | 1209 | 1825 | 1972 | 2794 | 3421 | ### Questions a) Does \( f^{-1}(t) \) exist? b) If \( f^{-1}(t) \) exists, what do the variables represent in the inverse function? c) If \( f^{-1}(t) \) exists, find \( f^{-1}(1825) \). d) If the table was extended to 2004 and if the factory sales of digital cameras for that year was $2794 million, would \( f^{-1}(t) \) exist? ### Explanation **a)** To determine if \( f^{-1}(t) \) exists, the function must be one-to-one. This means each \( f(t) \) should map to a unique year \( t \). **b)** If \( f^{-1}(t) \) exists, the variables would represent: \( f^{-1}(s) = t \), where \( s \) (sales value in millions) is mapped to \( t \) (the corresponding year). **c)** Looking up 1825 in the sales column of the table, the corresponding year \( t \) is 10. Thus, \( f^{-1}(1825) = 10 \). **d)** If a sales value is repeated for a different year, the function would not be one-to-one, and \( f^{-1}(t) \) would not exist. Since 2794 already corresponds to \( t = 12 \), having 2794 for an extended year would imply the inverse does not exist.
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