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functions

functions. mappings. example: a function. 10. 20. 30. 40. 50. 60. A man drops a ball from the top of a building. 10. 20. 30. 40. 50. 60. After ½ second, the ball has fallen 4 feet. Time. Distance. 1/2. 4. 10. 10. 20. 20. 30. 30. 40. 40. 50. 50. 60. 60.

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functions

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  1. functions mappings

  2. example: a function

  3. 10 20 30 40 50 60 A man drops a ball from the top of a building.

  4. 10 20 30 40 50 60 After ½ second, the ball has fallen 4 feet. Time Distance 1/2 4

  5. 10 10 20 20 30 30 40 40 50 50 60 60 After 1 second, the ball has fallen 16 feet. Time Distance 1/2 4

  6. After 3/2 second, the ball has fallen 36 feet. 10 10 10 20 20 20 30 30 30 40 40 40 50 50 50 60 60 60 Time Distance 1/2 4 3/2

  7. After 2 seconds, the ball has fallen 64 feet. 10 10 10 10 20 20 20 20 30 30 30 30 40 40 40 40 50 50 50 50 60 60 60 60 Time Distance 1/2 4 3/2

  8. 10 20 30 40 50 60 Motion is described as a set of ordered pairs. { ( 1/2 , 4 ), ( 1 ,16 ), ( 3/2 , 36 ), ( 2 , 64 ) } Time Distance 1/2 4 3/2

  9. 10 20 30 40 50 60 Motion is described as a set of ordered pairs. { ( 1/2 , 4 ), ( 1 ,16 ), ( 3/2 , 36 ), ( 2 , 64 ) } Time Distance 1/2 4 Sometimes there is a pattern, and we can write an equation: d = 16 t2 3/2 t is time in seconds d is distance in feet

  10. More generally, a function is defined as a set of ordered pairs. { ( 1/2 , 4 ), ( 1 ,16 ), ( 3/2 , 36 ), ( 2 , 64 ) } Time Distance 10 1/2 4 20 When we write an equation for a function, the solutions (ordered pairs) define the function. d = 16 t2 30 40 3/2 50 t is time in seconds d is distance in feet 60

  11. A function is defined as a set of ordered pairs. { ( 1/2 , 4 ), ( 1 ,16 ), ( 3/2 , 36 ), ( 2 , 64 ) } Time Distance 10 1/2 4 20 The DOMAIN of the function = {1/2 , 1 , 3/2 , 2} 30 40 The RANGE of the function = {4 ,16 , 36 , 64 } 3/2 50 60

  12. The DOMAIN of the function = 1/2 1 3/2 2 The RANGE of the function = 4 16 36 64 Time Distance 10 1/2 4 20 The function is a mapping that relates every Domain element t to a unique corresponding Range element, denoted f(t) and called the image of t 30 40 3/2 50 60

  13. The DOMAIN of the function = 1/2 1 3/2 2 The RANGE of the function = 4 16 36 64 Time Distance 10 1/2 4 20 The function is a mapping that relates every Domain element t to a unique corresponding Range element, denoted f(t) and called the image of t 30 40 4 is the image of ½ 16 is the image of 1 36 is the image of 3/2 64 is the image of 2 4 = f ( ½ ) 16 = f ( 1 ) 36 = f ( 3/2 ) 64 = f ( 2 ) 3/2 50 60

  14. The DOMAIN of the function = 1/2 1 3/2 2 The RANGE of the function = 4 16 36 64 Time Distance 10 1/2 4 20 The function (of high school algebra fame) relates a set of real numbers to another set of real numbers. Next we will examine a mapping that links a set of vectors to another set of vectors. In doing so, we use much of the same terminology that we used in the study of functions. A function is a type of mapping. 30 40 3/2 50 60

  15. example : a vector space mapping

  16. A farmer plans to purchase a herd of cows. He considers 2 breeds: Purple cows and Brown cows

  17. Each day a purple cow will eat 1 bale of hay and will produce 2 bottles of milk Each day a brown cow will eat 2 bales of hay and will produce 3 bottles of milk

  18. purple brown A herd comprised of 50 purple and 70 brown cows will consume 190 bales of hay and produce 310 bottles of milk.

  19. purple brown A herd comprised of 100 purple and 30 brown cows will consume 160 bales of hay and produce 290 bottles of milk.

  20. purple brown A herd comprised of 80 purple and 150 brown cows will consume 380 bales of hay and produce 610 bottles of milk.

  21. purple brown The DOMAIN of the mapping: These vectors describe the composition of the herd, and this determines

  22. purple brown The DOMAIN of the mapping: These vectors describe the composition of the herd, and this determines The RANGE of the mapping: These vectors describe the daily food intake and milk yield.

  23. purple brown Just as it is sometimes possible to find an equation to define a function, it is sometimes possible to produce a matrix to define a vector space mapping.

  24. purple brown Just as it is sometimes possible to find an equation to define a function, it is sometimes possible to produce a matrix to define a vector space mapping. # bales of hay = 1 (# purple cows) + 2 (# brown cows)

  25. purple brown Just as it is sometimes possible to find an equation to define a function, it is sometimes possible to produce a matrix to define a vector space mapping. # bales of hay = 1 (# purple cows) + 2 (# brown cows) # bottles of milk = 2 (# purple cows) + 3 (# brown cows)

  26. purple brown Just as it is sometimes possible to find an equation to define a function, it is sometimes possible to produce a matrix to define a vector space mapping. # bales of hay = 1 (# purple cows) + 2 (# brown cows) # bottles of milk = 2 (# purple cows) + 3 (# brown cows)

  27. purple brown eg:

  28. purple brown eg:

  29. purple brown For every domain element ( a vector in R2 whose entries are the numbers of each breed of cow) there is a unique corresponding range element ( a vector in R2 whose entries are the numbers of bales consumed and bottles produced.) A

  30. purple brown eg: A

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