Relations and Functions (Algebra 2)

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Students learn about relations and functions, and the vertical line test.
Students also learn to evaluate functions

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Relations and Functions (Algebra 2)

  1. 1. Relations and Functions  Analyze and graph relations.  Find functional values.1) ordered pair 8) function2) Cartesian Coordinate 9) mapping3) plane 10) one-to-one function4) quadrant 11) vertical line test5) relation 12) independent variable6) domain 13) dependent variable7) range 14) functional notation
  2. 2. Relations and FunctionsThis table shows the average lifetime Average Maximumand maximum lifetime for some animals. Animal Lifetime Lifetime (years) (years) Cat 12 28 Cow 15 30 Deer 8 20 Dog 12 20 Horse 20 50
  3. 3. Relations and FunctionsThis table shows the average lifetime Average Maximumand maximum lifetime for some animals. Animal Lifetime Lifetime (years) (years)The data can also be represented asordered pairs. Cat 12 28 Cow 15 30 Deer 8 20 Dog 12 20 Horse 20 50
  4. 4. Relations and FunctionsThis table shows the average lifetime Average Maximumand maximum lifetime for some animals. Animal Lifetime Lifetime (years) (years)The data can also be represented asordered pairs. Cat 12 28The ordered pairs for the data are: Cow 15 30 Deer 8 20 Dog 12 20 Horse 20 50
  5. 5. Relations and FunctionsThis table shows the average lifetime Average Maximumand maximum lifetime for some animals. Animal Lifetime Lifetime (years) (years)The data can also be represented asordered pairs. Cat 12 28The ordered pairs for the data are: Cow 15 30 (12, 28), (15, 30), (8, 20), (12, 20), and (20, 50) Deer 8 20 Dog 12 20 Horse 20 50
  6. 6. Relations and FunctionsThis table shows the average lifetime Average Maximumand maximum lifetime for some animals. Animal Lifetime Lifetime (years) (years)The data can also be represented asordered pairs. Cat 12 28The ordered pairs for the data are: Cow 15 30 (12, 28), (15, 30), (8, 20), (12, 20), and (20, 50) Deer 8 20The first number in each ordered pair Dog 12 20is the average lifetime, and the secondnumber is the maximum lifetime. Horse 20 50
  7. 7. Relations and FunctionsThis table shows the average lifetime Average Maximumand maximum lifetime for some animals. Animal Lifetime Lifetime (years) (years)The data can also be represented asordered pairs. Cat 12 28The ordered pairs for the data are: Cow 15 30 (12, 28), (15, 30), (8, 20), (12, 20), and (20, 50) Deer 8 20The first number in each ordered pair Dog 12 20is the average lifetime, and the secondnumber is the maximum lifetime. Horse 20 50 (20, 50)
  8. 8. Relations and FunctionsThis table shows the average lifetime Average Maximumand maximum lifetime for some animals. Animal Lifetime Lifetime (years) (years)The data can also be represented asordered pairs. Cat 12 28The ordered pairs for the data are: Cow 15 30 (12, 28), (15, 30), (8, 20), (12, 20), and (20, 50) Deer 8 20The first number in each ordered pair Dog 12 20is the average lifetime, and the secondnumber is the maximum lifetime. Horse 20 50 (20, 50) average lifetime
  9. 9. Relations and FunctionsThis table shows the average lifetime Average Maximumand maximum lifetime for some animals. Animal Lifetime Lifetime (years) (years)The data can also be represented asordered pairs. Cat 12 28The ordered pairs for the data are: Cow 15 30 (12, 28), (15, 30), (8, 20), (12, 20), and (20, 50) Deer 8 20The first number in each ordered pair Dog 12 20is the average lifetime, and the secondnumber is the maximum lifetime. Horse 20 50 (20, 50) average maximum lifetime lifetime
  10. 10. Relations and FunctionsYou can graph the ordered pairs below Animal Lifetimeson a coordinate system with two axes. y 60 50 Maximum Lifetime 40 30 20 10 0 x 0 5 10 15 20 25 30 Average Lifetime
  11. 11. Relations and FunctionsYou can graph the ordered pairs below Animal Lifetimeson a coordinate system with two axes. y (12, 28), 60 50 Maximum Lifetime 40 30 20 10 0 x 0 5 10 15 20 25 30 Average Lifetime
  12. 12. Relations and FunctionsYou can graph the ordered pairs below Animal Lifetimeson a coordinate system with two axes. y (12, 28), (15, 30), 60 50 Maximum Lifetime 40 30 20 10 0 x 0 5 10 15 20 25 30 Average Lifetime
  13. 13. Relations and FunctionsYou can graph the ordered pairs below Animal Lifetimeson a coordinate system with two axes. y (12, 28), (15, 30), (8, 20), 60 50 Maximum Lifetime 40 30 20 10 0 x 0 5 10 15 20 25 30 Average Lifetime
  14. 14. Relations and FunctionsYou can graph the ordered pairs below Animal Lifetimeson a coordinate system with two axes. y (12, 28), (15, 30), (8, 20), 60 (12, 20), 50 Maximum Lifetime 40 30 20 10 0 x 0 5 10 15 20 25 30 Average Lifetime
  15. 15. Relations and FunctionsYou can graph the ordered pairs below Animal Lifetimeson a coordinate system with two axes. y (12, 28), (15, 30), (8, 20), 60 (12, 20), and (20, 50) 50 Maximum Lifetime 40 30 20 10 0 x 0 5 10 15 20 25 30 Average Lifetime
  16. 16. Relations and FunctionsYou can graph the ordered pairs below Animal Lifetimeson a coordinate system with two axes. y (12, 28), (15, 30), (8, 20), 60 (12, 20), and (20, 50) 50 Maximum Lifetime 40Remember, each point in the coordinateplane can be named by exactly one 30ordered pair and that every ordered pairnames exactly one point in the coordinate 20plane. 10 0 x 0 5 10 15 20 25 30 Average Lifetime
  17. 17. Relations and FunctionsYou can graph the ordered pairs below Animal Lifetimeson a coordinate system with two axes. y (12, 28), (15, 30), (8, 20), 60 (12, 20), and (20, 50) 50 Maximum Lifetime 40Remember, each point in the coordinateplane can be named by exactly one 30ordered pair and that every ordered pairnames exactly one point in the coordinate 20plane. 10The graph of this data (animal lifetimes) 0 xlies in only one part of the Cartesian 0 5 10 15 20 25 30coordinate plane – the part with all Average Lifetimepositive numbers.
  18. 18. Relations and FunctionsThe Cartesian coordinate system is composed of the x-axis (horizontal), -5 0 5
  19. 19. Relations and FunctionsThe Cartesian coordinate system is composed of the x-axis (horizontal),and the y-axis (vertical), which meet at the origin (0, 0) and divide the plane intofour quadrants. 5 Origin (0, 0) 0 -5 0 5 -5
  20. 20. Relations and FunctionsThe Cartesian coordinate system is composed of the x-axis (horizontal),and the y-axis (vertical), which meet at the origin (0, 0) and divide the plane intofour quadrants.You can tell which quadrant a point is in by looking at the sign of each coordinate ofthe point. 5 Quadrant II Origin Quadrant I ( --, + ) ( +, (0, )0) + 0 -5 0 5 Quadrant III Quadrant IV ( --, -- ) ( +, -- ) -5
  21. 21. Relations and FunctionsThe Cartesian coordinate system is composed of the x-axis (horizontal),and the y-axis (vertical), which meet at the origin (0, 0) and divide the plane intofour quadrants.You can tell which quadrant a point is in by looking at the sign of each coordinate ofthe point. 5 Quadrant II Origin Quadrant I ( --, + ) ( +, (0, )0) + 0 -5 0 5 Quadrant III Quadrant IV ( --, -- ) ( +, -- ) -5 The points on the two axes do not lie in any quadrant.
  22. 22. Relations and FunctionsIn general, any ordered pair in the coordinate plane can be written in the form (x, y)
  23. 23. Relations and FunctionsIn general, any ordered pair in the coordinate plane can be written in the form (x, y)A relation is a set of ordered pairs, such as the one for the longevity of animals.
  24. 24. Relations and FunctionsIn general, any ordered pair in the coordinate plane can be written in the form (x, y)A relation is a set of ordered pairs, such as the one for the longevity of animals.The domain of a relation is the set of all first coordinates (x-coordinates) from theordered pairs.
  25. 25. Relations and FunctionsIn general, any ordered pair in the coordinate plane can be written in the form (x, y)A relation is a set of ordered pairs, such as the one for the longevity of animals.The domain of a relation is the set of all first coordinates (x-coordinates) from theordered pairs.The range of a relation is the set of all second coordinates (y-coordinates) from theordered pairs.
  26. 26. Relations and FunctionsIn general, any ordered pair in the coordinate plane can be written in the form (x, y)A relation is a set of ordered pairs, such as the one for the longevity of animals.The domain of a relation is the set of all first coordinates (x-coordinates) from theordered pairs.The range of a relation is the set of all second coordinates (y-coordinates) from theordered pairs.The graph of a relation is the set of points in the coordinate plane corresponding to theordered pairs in the relation.
  27. 27. Relations and FunctionsA function is a special type of relation in which each element of the domain is pairedwith ___________ element in the range.
  28. 28. Relations and FunctionsA function is a special type of relation in which each element of the domain is pairedwith ___________ element in the range. exactly one
  29. 29. Relations and FunctionsA function is a special type of relation in which each element of the domain is pairedwith ___________ element in the range. exactly oneA mapping shows how each member of the domain is paired with each member inthe range.
  30. 30. Relations and FunctionsA function is a special type of relation in which each element of the domain is pairedwith ___________ element in the range. exactly oneA mapping shows how each member of the domain is paired with each member inthe range. Functions    3,1 ,  0,2 ,  2,4
  31. 31. Relations and FunctionsA function is a special type of relation in which each element of the domain is pairedwith ___________ element in the range. exactly oneA mapping shows how each member of the domain is paired with each member inthe range. Functions    3,1 ,  0,2 ,  2,4 Domain -3 0 2
  32. 32. Relations and FunctionsA function is a special type of relation in which each element of the domain is pairedwith ___________ element in the range. exactly oneA mapping shows how each member of the domain is paired with each member inthe range. Functions    3,1 ,  0,2 ,  2,4 Domain Range -3 1 0 2 2 4
  33. 33. Relations and FunctionsA function is a special type of relation in which each element of the domain is pairedwith ___________ element in the range. exactly oneA mapping shows how each member of the domain is paired with each member inthe range. Functions    3,1 ,  0,2 ,  2,4 Domain Range -3 1 0 2 2 4
  34. 34. Relations and FunctionsA function is a special type of relation in which each element of the domain is pairedwith ___________ element in the range. exactly oneA mapping shows how each member of the domain is paired with each member inthe range. Functions    3,1 ,  0,2 ,  2,4 Domain Range -3 1 0 2 2 4
  35. 35. Relations and FunctionsA function is a special type of relation in which each element of the domain is pairedwith ___________ element in the range. exactly oneA mapping shows how each member of the domain is paired with each member inthe range. Functions    3,1 ,  0,2 ,  2,4 Domain Range -3 1 0 2 2 4
  36. 36. Relations and FunctionsA function is a special type of relation in which each element of the domain is pairedwith ___________ element in the range. exactly oneA mapping shows how each member of the domain is paired with each member inthe range. Functions    3,1 ,  0,2 ,  2,4 Domain Range -3 1 0 2 2 4 one-to-one function
  37. 37. Relations and FunctionsA function is a special type of relation in which each element of the domain is pairedwith ___________ element in the range. exactly oneA mapping shows how each member of the domain is paired with each member inthe range. Functions    1,5 , 1,3 ,  4,5
  38. 38. Relations and FunctionsA function is a special type of relation in which each element of the domain is pairedwith ___________ element in the range. exactly oneA mapping shows how each member of the domain is paired with each member inthe range. Functions    1,5 , 1,3 ,  4,5 Domain Range -1 5 1 3 4
  39. 39. Relations and FunctionsA function is a special type of relation in which each element of the domain is pairedwith ___________ element in the range. exactly oneA mapping shows how each member of the domain is paired with each member inthe range. Functions    1,5 , 1,3 ,  4,5 Domain Range -1 5 1 3 4
  40. 40. Relations and FunctionsA function is a special type of relation in which each element of the domain is pairedwith ___________ element in the range. exactly oneA mapping shows how each member of the domain is paired with each member inthe range. Functions    1,5 , 1,3 ,  4,5 Domain Range -1 5 1 3 4
  41. 41. Relations and FunctionsA function is a special type of relation in which each element of the domain is pairedwith ___________ element in the range. exactly oneA mapping shows how each member of the domain is paired with each member inthe range. Functions    1,5 , 1,3 ,  4,5 Domain Range -1 5 1 3 4
  42. 42. Relations and FunctionsA function is a special type of relation in which each element of the domain is pairedwith ___________ element in the range. exactly oneA mapping shows how each member of the domain is paired with each member inthe range. Functions    1,5 , 1,3 ,  4,5 Domain Range -1 5 1 3 4 function, not one-to-one
  43. 43. Relations and FunctionsA function is a special type of relation in which each element of the domain is pairedwith ___________ element in the range. exactly oneA mapping shows how each member of the domain is paired with each member inthe range. Functions   5,6 ,   3,0 , 1,1 ,   3,6
  44. 44. Relations and FunctionsA function is a special type of relation in which each element of the domain is pairedwith ___________ element in the range. exactly oneA mapping shows how each member of the domain is paired with each member inthe range. Functions   5,6 ,   3,0 , 1,1 ,   3,6 Domain Range 5 6 -3 0 1 1
  45. 45. Relations and FunctionsA function is a special type of relation in which each element of the domain is pairedwith ___________ element in the range. exactly oneA mapping shows how each member of the domain is paired with each member inthe range. Functions   5,6 ,   3,0 , 1,1 ,   3,6 Domain Range 5 6 -3 0 1 1
  46. 46. Relations and FunctionsA function is a special type of relation in which each element of the domain is pairedwith ___________ element in the range. exactly oneA mapping shows how each member of the domain is paired with each member inthe range. Functions   5,6 ,   3,0 , 1,1 ,   3,6 Domain Range 5 6 -3 0 1 1
  47. 47. Relations and FunctionsA function is a special type of relation in which each element of the domain is pairedwith ___________ element in the range. exactly oneA mapping shows how each member of the domain is paired with each member inthe range. Functions   5,6 ,   3,0 , 1,1 ,   3,6 Domain Range 5 6 -3 0 1 1 not a function
  48. 48. Relations and FunctionsA function is a special type of relation in which each element of the domain is pairedwith ___________ element in the range. exactly oneA mapping shows how each member of the domain is paired with each member inthe range. Functions   5,6 ,   3,0 , 1,1 ,   3,6 Domain Range 5 6 -3 0 1 1 not a function
  49. 49. Relations and FunctionsState the domain and range of the relation shown yin the graph. Is the relation a function? (-4,3) (2,3) x (-1,-2) (3,-3) (0,-4)
  50. 50. Relations and FunctionsState the domain and range of the relation shown yin the graph. Is the relation a function? (-4,3) (2,3)The relation is: { (-4, 3), (-1, 2), (0, -4), (2, 3), (3, -3) } x (-1,-2) (3,-3) (0,-4)
  51. 51. Relations and FunctionsState the domain and range of the relation shown yin the graph. Is the relation a function? (-4,3) (2,3)The relation is: { (-4, 3), (-1, 2), (0, -4), (2, 3), (3, -3) }The domain is: x (-1,-2) (3,-3) (0,-4)
  52. 52. Relations and FunctionsState the domain and range of the relation shown yin the graph. Is the relation a function? (-4,3) (2,3)The relation is: { (-4, 3), (-1, 2), (0, -4), (2, 3), (3, -3) }The domain is: x { -4, -1, 0, 2, 3 } (-1,-2) (3,-3) (0,-4)
  53. 53. Relations and FunctionsState the domain and range of the relation shown yin the graph. Is the relation a function? (-4,3) (2,3)The relation is: { (-4, 3), (-1, 2), (0, -4), (2, 3), (3, -3) }The domain is: x { -4, -1, 0, 2, 3 } (-1,-2) (3,-3)The range is: (0,-4)
  54. 54. Relations and FunctionsState the domain and range of the relation shown yin the graph. Is the relation a function? (-4,3) (2,3)The relation is: { (-4, 3), (-1, 2), (0, -4), (2, 3), (3, -3) }The domain is: x { -4, -1, 0, 2, 3 } (-1,-2) (3,-3)The range is: (0,-4) { -4, -3, -2, 3 }
  55. 55. Relations and FunctionsState the domain and range of the relation shown yin the graph. Is the relation a function? (-4,3) (2,3)The relation is: { (-4, 3), (-1, 2), (0, -4), (2, 3), (3, -3) }The domain is: x { -4, -1, 0, 2, 3 } (-1,-2) (3,-3)The range is: (0,-4) { -4, -3, -2, 3 }Each member of the domain is paired with exactly one member of the range,so this relation is a function.
  56. 56. Relations and FunctionsYou can use the vertical line test to determine whether a relation is a function.
  57. 57. Relations and FunctionsYou can use the vertical line test to determine whether a relation is a function. Vertical Line Test If no vertical line intersects a graph in more than one point, the graph represents a function. y x
  58. 58. Relations and FunctionsYou can use the vertical line test to determine whether a relation is a function. Vertical Line Test If no vertical line intersects a graph in more than one point, the graph represents a function. y x
  59. 59. Relations and FunctionsYou can use the vertical line test to determine whether a relation is a function. Vertical Line Test If no vertical line intersects a graph in more than one point, the graph represents a function. y x
  60. 60. Relations and FunctionsYou can use the vertical line test to determine whether a relation is a function. Vertical Line Test If no vertical line intersects a graph in more than one point, the graph represents a function. y x
  61. 61. Relations and FunctionsYou can use the vertical line test to determine whether a relation is a function. Vertical Line Test If no vertical line intersects a graph in more than one point, the graph represents a function. y x
  62. 62. Relations and FunctionsYou can use the vertical line test to determine whether a relation is a function. Vertical Line Test If no vertical line intersects a graph in more than one point, the graph represents a function. y x
  63. 63. Relations and FunctionsYou can use the vertical line test to determine whether a relation is a function. Vertical Line Test If no vertical line intersects a graph in more than one point, the graph represents a function. y x
  64. 64. Relations and FunctionsYou can use the vertical line test to determine whether a relation is a function. Vertical Line Test If no vertical line intersects a graph in more than one point, the graph represents a function. y x
  65. 65. Relations and FunctionsYou can use the vertical line test to determine whether a relation is a function. Vertical Line Test If no vertical line intersects a If some vertical line intercepts a graph in more than one point, graph in two or more points, the the graph represents a function. graph does not represent a function. y y x x
  66. 66. Relations and FunctionsYou can use the vertical line test to determine whether a relation is a function. Vertical Line Test If no vertical line intersects a If some vertical line intercepts a graph in more than one point, graph in two or more points, the the graph represents a function. graph does not represent a function. y y x x
  67. 67. Relations and FunctionsYou can use the vertical line test to determine whether a relation is a function. Vertical Line Test If no vertical line intersects a If some vertical line intercepts a graph in more than one point, graph in two or more points, the the graph represents a function. graph does not represent a function. y y x x
  68. 68. Relations and FunctionsYou can use the vertical line test to determine whether a relation is a function. Vertical Line Test If no vertical line intersects a If some vertical line intercepts a graph in more than one point, graph in two or more points, the the graph represents a function. graph does not represent a function. y y x x
  69. 69. Relations and FunctionsThe table shows the population of Indiana over the last several Population Yeardecades. (millions) 1950 3.9 1960 4.7 1970 5.2 1980 5.5 1990 5.5 2000 6.1
  70. 70. Relations and FunctionsThe table shows the population of Indiana over the last several Population Yeardecades. (millions) 1950 3.9We can graph this data to determine 1960 4.7if it represents a function. 1970 5.2 1980 5.5 1990 5.5 2000 6.1
  71. 71. Relations and FunctionsThe table shows the population of Indiana over the last several Population Yeardecades. (millions) 1950 3.9We can graph this data to determine 1960 4.7if it represents a function. 1970 5.2 Population of Indiana 8 1980 5.5 7 6 1990 5.5 Population 5 (millions) 2000 6.1 4 3 2 1 0 ‘50 ‘60 ‘70 ‘80 ‘90 ‘00 7 0 Year
  72. 72. Relations and FunctionsThe table shows the population of Indiana over the last several Population Yeardecades. (millions) 1950 3.9We can graph this data to determine 1960 4.7if it represents a function. 1970 5.2 Population of Indiana 8 1980 5.5 7 6 1990 5.5 Use the vertical Population 5 (millions) line test. 2000 6.1 4 3 2 1 0 ‘50 ‘60 ‘70 ‘80 ‘90 ‘00 7 0 Year
  73. 73. Relations and FunctionsThe table shows the population of Indiana over the last several Population Yeardecades. (millions) 1950 3.9We can graph this data to determine 1960 4.7if it represents a function. 1970 5.2 Population of Indiana 8 1980 5.5 7 6 1990 5.5 Use the vertical Population 5 (millions) line test. 2000 6.1 4 3 2 1 0 ‘50 ‘60 ‘70 ‘80 ‘90 ‘00 7 0 Year
  74. 74. Relations and FunctionsThe table shows the population of Indiana over the last several Population Yeardecades. (millions) 1950 3.9We can graph this data to determine 1960 4.7if it represents a function. 1970 5.2 Population of Indiana 8 1980 5.5 7 6 1990 5.5 Use the vertical Population 5 (millions) line test. 2000 6.1 4 3 2 1 0 ‘50 ‘60 ‘70 ‘80 ‘90 ‘00 7 0 Year
  75. 75. Relations and FunctionsThe table shows the population of Indiana over the last several Population Yeardecades. (millions) 1950 3.9We can graph this data to determine 1960 4.7if it represents a function. 1970 5.2 Population of Indiana 8 1980 5.5 7 6 1990 5.5 Use the vertical Population 5 (millions) line test. 2000 6.1 4 3 2 1 0 ‘50 ‘60 ‘70 ‘80 ‘90 ‘00 7 0 Year
  76. 76. Relations and FunctionsThe table shows the population of Indiana over the last several Population Yeardecades. (millions) 1950 3.9We can graph this data to determine 1960 4.7if it represents a function. 1970 5.2 Population of Indiana 8 1980 5.5 7 6 1990 5.5 Use the vertical Population 5 (millions) line test. 2000 6.1 4 3 2 1 0 ‘50 ‘60 ‘70 ‘80 ‘90 ‘00 7 0 Year
  77. 77. Relations and FunctionsThe table shows the population of Indiana over the last several Population Yeardecades. (millions) 1950 3.9We can graph this data to determine 1960 4.7if it represents a function. 1970 5.2 Population of Indiana 8 1980 5.5 7 6 1990 5.5 Use the vertical Population 5 (millions) line test. 2000 6.1 4 3 2 1 0 ‘50 ‘60 ‘70 ‘80 ‘90 ‘00 7 0 Year
  78. 78. Relations and FunctionsThe table shows the population of Indiana over the last several Population Yeardecades. (millions) 1950 3.9We can graph this data to determine 1960 4.7if it represents a function. 1970 5.2 Population of Indiana 8 1980 5.5 7 6 1990 5.5 Use the vertical Population 5 (millions) line test. 2000 6.1 4 3 2 1 0 ‘50 ‘60 ‘70 ‘80 ‘90 ‘00 7 0 Year
  79. 79. Relations and FunctionsThe table shows the population of Indiana over the last several Population Yeardecades. (millions) 1950 3.9We can graph this data to determine 1960 4.7if it represents a function. 1970 5.2 Population of Indiana 8 1980 5.5 7 6 1990 5.5 Use the vertical Population 5 (millions) line test. 2000 6.1 4 3 2 1 0 ‘50 ‘60 ‘70 ‘80 ‘90 ‘00 7 0 Year
  80. 80. Relations and FunctionsThe table shows the population of Indiana over the last several Population Yeardecades. (millions) 1950 3.9We can graph this data to determine 1960 4.7if it represents a function. 1970 5.2 Population of Indiana 8 1980 5.5 7 6 1990 5.5 Use the vertical Population 5 (millions) line test. 2000 6.1 4 3 2 1 0 ‘50 ‘60 ‘70 ‘80 ‘90 ‘00 7 0 Year
  81. 81. Relations and FunctionsThe table shows the population of Indiana over the last several Population Yeardecades. (millions) 1950 3.9We can graph this data to determine 1960 4.7if it represents a function. 1970 5.2 Population of Indiana 8 1980 5.5 7 6 1990 5.5 Use the vertical Population 5 (millions) line test. 2000 6.1 4 3 Notice that no vertical line can be drawn that 2 contains more than one of the data points. 1 0 ‘50 ‘60 ‘70 ‘80 ‘90 ‘00 7 0 Year
  82. 82. Relations and FunctionsThe table shows the population of Indiana over the last several Population Yeardecades. (millions) 1950 3.9We can graph this data to determine 1960 4.7if it represents a function. 1970 5.2 Population of Indiana 8 1980 5.5 7 6 1990 5.5 Use the vertical Population 5 (millions) line test. 2000 6.1 4 3 Notice that no vertical line can be drawn that 2 contains more than one of the data points. 1 0 Therefore, this relation is a function! ‘50 ‘60 ‘70 ‘80 ‘90 ‘00 7 0 Year
  83. 83. Relations and FunctionsGraph the relation y  2 x  1
  84. 84. Relations and FunctionsGraph the relation y  2 x  11) Make a table of values.
  85. 85. Relations and FunctionsGraph the relation y  2 x  11) Make a table of values. x y -1 -1 0 1 1 3 2 5
  86. 86. Relations and FunctionsGraph the relation y  2 x  1 2) Graph the ordered pairs.1) Make a table of values. x y -1 -1 0 1 1 3 2 5
  87. 87. Relations and FunctionsGraph the relation y  2 x  1 2) Graph the ordered pairs.1) Make a table of values. y 7 6 x y 5 4 -1 -1 3 2 0 1 1 0 x 1 3 -1 2 5 -2 -3 -5 -4 -3 -2 -1 1 2 3 4 5 0
  88. 88. Relations and FunctionsGraph the relation y  2 x  1 2) Graph the ordered pairs.1) Make a table of values. y 7 6 x y 5 4 -1 -1 3 2 0 1 1 0 x 1 3 -1 2 5 -2 -3 -5 -4 -3 -2 -1 1 2 3 4 5 03) Find the domain and range.
  89. 89. Relations and FunctionsGraph the relation y  2 x  1 2) Graph the ordered pairs.1) Make a table of values. y 7 6 x y 5 4 -1 -1 3 2 0 1 1 0 x 1 3 -1 2 5 -2 -3 -5 -4 -3 -2 -1 1 2 3 4 5 03) Find the domain and range. Domain is all real numbers.
  90. 90. Relations and FunctionsGraph the relation y  2 x  1 2) Graph the ordered pairs.1) Make a table of values. y 7 6 x y 5 4 -1 -1 3 2 0 1 1 0 x 1 3 -1 2 5 -2 -3 -5 -4 -3 -2 -1 1 2 3 4 5 03) Find the domain and range. Domain is all real numbers. Range is all real numbers.
  91. 91. Relations and FunctionsGraph the relation y  2 x  1 2) Graph the ordered pairs.1) Make a table of values. y 7 6 x y 5 4 -1 -1 3 2 0 1 1 0 x 1 3 -1 2 5 -2 -3 -5 -4 -3 -2 -1 1 2 3 4 5 03) Find the domain and range. 4) Determine whether the relation is a function. Domain is all real numbers. Range is all real numbers.
  92. 92. Relations and FunctionsGraph the relation y  2 x  1 2) Graph the ordered pairs.1) Make a table of values. y 7 6 x y 5 4 -1 -1 3 2 0 1 1 0 x 1 3 -1 2 5 -2 -3 -5 -4 -3 -2 -1 1 2 3 4 5 03) Find the domain and range. 4) Determine whether the relation is a function. Domain is all real numbers. Range is all real numbers.
  93. 93. Relations and FunctionsGraph the relation y  2 x  1 2) Graph the ordered pairs.1) Make a table of values. y 7 6 x y 5 4 -1 -1 3 2 0 1 1 0 x 1 3 -1 2 5 -2 -3 -5 -4 -3 -2 -1 1 2 3 4 5 03) Find the domain and range. 4) Determine whether the relation is a function. Domain is all real numbers. Range is all real numbers.
  94. 94. Relations and FunctionsGraph the relation y  2 x  1 2) Graph the ordered pairs.1) Make a table of values. y 7 6 x y 5 4 -1 -1 3 2 0 1 1 0 x 1 3 -1 2 5 -2 -3 -5 -4 -3 -2 -1 1 2 3 4 5 03) Find the domain and range. 4) Determine whether the relation is a function. Domain is all real numbers. Range is all real numbers.
  95. 95. Relations and FunctionsGraph the relation y  2 x  1 2) Graph the ordered pairs.1) Make a table of values. y 7 6 x y 5 4 -1 -1 3 2 0 1 1 0 x 1 3 -1 2 5 -2 -3 -5 -4 -3 -2 -1 1 2 3 4 5 03) Find the domain and range. 4) Determine whether the relation is a function. Domain is all real numbers. Range is all real numbers.
  96. 96. Relations and FunctionsGraph the relation y  2 x  1 2) Graph the ordered pairs.1) Make a table of values. y 7 6 x y 5 4 -1 -1 3 2 0 1 1 0 x 1 3 -1 2 5 -2 -3 -5 -4 -3 -2 -1 1 2 3 4 5 03) Find the domain and range. 4) Determine whether the relation is a function. Domain is all real numbers. The graph passes the vertical line test. Range is all real numbers.
  97. 97. Relations and FunctionsGraph the relation y  2 x  1 2) Graph the ordered pairs.1) Make a table of values. y 7 6 x y 5 4 -1 -1 3 2 0 1 1 0 x 1 3 -1 2 5 -2 -3 -5 -4 -3 -2 -1 1 2 3 4 5 03) Find the domain and range. 4) Determine whether the relation is a function. Domain is all real numbers. The graph passes the vertical line test. Range is all real numbers. For every x value there is exactly one y value, so the equation y = 2x + 1 represents a function.
  98. 98. Relations and FunctionsGraph the relation x  y 2  2
  99. 99. Relations and FunctionsGraph the relation x  y 2  21) Make a table of values.
  100. 100. Relations and FunctionsGraph the relation x  y 2  21) Make a table of values. x y 2 -2 -1 -1 -2 0 -1 1 2 2
  101. 101. Relations and FunctionsGraph the relation x  y 2  2 2) Graph the ordered pairs.1) Make a table of values. x y 2 -2 -1 -1 -2 0 -1 1 2 2
  102. 102. Relations and FunctionsGraph the relation x  y 2  2 2) Graph the ordered pairs. y1) Make a table of values. 7 6 5 x y 4 2 -2 3 2 -1 -1 1 0 x -2 0 -1 -2 -1 1 -3 -5 -4 -3 -2 -1 1 2 3 4 5 0 2 2
  103. 103. Relations and FunctionsGraph the relation x  y 2  2 2) Graph the ordered pairs. y1) Make a table of values. 7 6 5 x y 4 2 -2 3 2 -1 -1 1 0 x -2 0 -1 -2 -1 1 -3 -5 -4 -3 -2 -1 1 2 3 4 5 0 2 23) Find the domain and range.
  104. 104. Relations and FunctionsGraph the relation x  y 2  2 2) Graph the ordered pairs. y1) Make a table of values. 7 6 5 x y 4 2 -2 3 2 -1 -1 1 0 x -2 0 -1 -2 -1 1 -3 -5 -4 -3 -2 -1 1 2 3 4 5 0 2 23) Find the domain and range. Domain is all real numbers, greater than or equal to -2.
  105. 105. Relations and FunctionsGraph the relation x  y 2  2 2) Graph the ordered pairs. y1) Make a table of values. 7 6 5 x y 4 2 -2 3 2 -1 -1 1 0 x -2 0 -1 -2 -1 1 -3 -5 -4 -3 -2 -1 1 2 3 4 5 0 2 23) Find the domain and range. Domain is all real numbers, greater than or equal to -2. Range is all real numbers.
  106. 106. Relations and FunctionsGraph the relation x  y 2  2 2) Graph the ordered pairs. y1) Make a table of values. 7 6 5 x y 4 2 -2 3 2 -1 -1 1 0 x -2 0 -1 -2 -1 1 -3 -5 -4 -3 -2 -1 1 2 3 4 5 0 2 2 4) Determine whether the relation is a function.3) Find the domain and range. Domain is all real numbers, greater than or equal to -2. Range is all real numbers.
  107. 107. Relations and FunctionsGraph the relation x  y 2  2 2) Graph the ordered pairs. y1) Make a table of values. 7 6 5 x y 4 2 -2 3 2 -1 -1 1 0 x -2 0 -1 -2 -1 1 -3 -5 -4 -3 -2 -1 1 2 3 4 5 0 2 2 4) Determine whether the relation is a function.3) Find the domain and range. Domain is all real numbers, greater than or equal to -2. Range is all real numbers.
  108. 108. Relations and FunctionsGraph the relation x  y 2  2 2) Graph the ordered pairs. y1) Make a table of values. 7 6 5 x y 4 2 -2 3 2 -1 -1 1 0 x -2 0 -1 -2 -1 1 -3 -5 -4 -3 -2 -1 1 2 3 4 5 0 2 2 4) Determine whether the relation is a function.3) Find the domain and range. Domain is all real numbers, greater than or equal to -2. Range is all real numbers.
  109. 109. Relations and FunctionsGraph the relation x  y 2  2 2) Graph the ordered pairs. y1) Make a table of values. 7 6 5 x y 4 2 -2 3 2 -1 -1 1 0 x -2 0 -1 -2 -1 1 -3 -5 -4 -3 -2 -1 1 2 3 4 5 0 2 2 4) Determine whether the relation is a function.3) Find the domain and range. The graph does not pass the vertical line test. Domain is all real numbers, greater than or equal to -2. Range is all real numbers.
  110. 110. Relations and FunctionsGraph the relation x  y 2  2 2) Graph the ordered pairs. y1) Make a table of values. 7 6 5 x y 4 2 -2 3 2 -1 -1 1 0 x -2 0 -1 -2 -1 1 -3 -5 -4 -3 -2 -1 1 2 3 4 5 0 2 2 4) Determine whether the relation is a function.3) Find the domain and range. The graph does not pass the vertical line test. Domain is all real numbers, For every x value (except x = -2), greater than or equal to -2. there are TWO y values, Range is all real numbers. so the equation x = y2 – 2 DOES NOT represent a function.
  111. 111. Relations and FunctionsWhen an equation represents a function, the variable (usually x) whose values makeup the domain is called the independent variable.
  112. 112. Relations and FunctionsWhen an equation represents a function, the variable (usually x) whose values makeup the domain is called the independent variable.The other variable (usually y) whose values make up the range is called thedependent variable because its values depend on x.
  113. 113. Relations and FunctionsWhen an equation represents a function, the variable (usually x) whose values makeup the domain is called the independent variable.The other variable (usually y) whose values make up the range is called thedependent variable because its values depend on x.Equations that represent functions are often written in function notation.
  114. 114. Relations and FunctionsWhen an equation represents a function, the variable (usually x) whose values makeup the domain is called the independent variable.The other variable (usually y) whose values make up the range is called thedependent variable because its values depend on x.Equations that represent functions are often written in function notation.The equation y = 2x + 1 can be written as f(x) = 2x + 1.
  115. 115. Relations and FunctionsWhen an equation represents a function, the variable (usually x) whose values makeup the domain is called the independent variable.The other variable (usually y) whose values make up the range is called thedependent variable because its values depend on x.Equations that represent functions are often written in function notation.The equation y = 2x + 1 can be written as f(x) = 2x + 1. yThe symbol f(x) replaces the __ , and is read “f of x”
  116. 116. Relations and FunctionsWhen an equation represents a function, the variable (usually x) whose values makeup the domain is called the independent variable.The other variable (usually y) whose values make up the range is called thedependent variable because its values depend on x.Equations that represent functions are often written in function notation.The equation y = 2x + 1 can be written as f(x) = 2x + 1. yThe symbol f(x) replaces the __ , and is read “f of x”The f is just the name of the function. It is NOT a variable that is multiplied by x.
  117. 117. Relations and FunctionsSuppose you want to find the value in the range that corresponds to the element4 in the domain of the function. f(x) = 2x + 1
  118. 118. Relations and FunctionsSuppose you want to find the value in the range that corresponds to the element4 in the domain of the function. f(x) = 2x + 1This is written as f(4) and is read “f of 4.”
  119. 119. Relations and FunctionsSuppose you want to find the value in the range that corresponds to the element4 in the domain of the function. f(x) = 2x + 1This is written as f(4) and is read “f of 4.”The value f(4) is found by substituting 4 for each x in the equation.
  120. 120. Relations and FunctionsSuppose you want to find the value in the range that corresponds to the element4 in the domain of the function. f(x) = 2x + 1This is written as f(4) and is read “f of 4.”The value f(4) is found by substituting 4 for each x in the equation. Therefore, if f(x) = 2x + 1
  121. 121. Relations and FunctionsSuppose you want to find the value in the range that corresponds to the element4 in the domain of the function. f(x) = 2x + 1This is written as f(4) and is read “f of 4.”The value f(4) is found by substituting 4 for each x in the equation. Therefore, if f(x) = 2x + 1 Then f(4) = 2(4) + 1
  122. 122. Relations and FunctionsSuppose you want to find the value in the range that corresponds to the element4 in the domain of the function. f(x) = 2x + 1This is written as f(4) and is read “f of 4.”The value f(4) is found by substituting 4 for each x in the equation. Therefore, if f(x) = 2x + 1 Then f(4) = 2(4) + 1 f(4) = 8 + 1
  123. 123. Relations and FunctionsSuppose you want to find the value in the range that corresponds to the element4 in the domain of the function. f(x) = 2x + 1This is written as f(4) and is read “f of 4.”The value f(4) is found by substituting 4 for each x in the equation. Therefore, if f(x) = 2x + 1 Then f(4) = 2(4) + 1 f(4) = 8 + 1 f(4) = 9
  124. 124. Relations and FunctionsSuppose you want to find the value in the range that corresponds to the element4 in the domain of the function. f(x) = 2x + 1This is written as f(4) and is read “f of 4.”The value f(4) is found by substituting 4 for each x in the equation. Therefore, if f(x) = 2x + 1 Then f(4) = 2(4) + 1 f(4) = 8 + 1 f(4) = 9NOTE: Letters other than f can be used to represent a function.
  125. 125. Relations and FunctionsSuppose you want to find the value in the range that corresponds to the element4 in the domain of the function. f(x) = 2x + 1This is written as f(4) and is read “f of 4.”The value f(4) is found by substituting 4 for each x in the equation. Therefore, if f(x) = 2x + 1 Then f(4) = 2(4) + 1 f(4) = 8 + 1 f(4) = 9NOTE: Letters other than f can be used to represent a function.EXAMPLE: g(x) = 2x + 1
  126. 126. Relations and FunctionsGiven: f(x) = x2 + 2 and g(x) = 0.5x2 – 5x + 3.5Find each value.
  127. 127. Relations and FunctionsGiven: f(x) = x2 + 2 and g(x) = 0.5x2 – 5x + 3.5Find each value. f(-3)
  128. 128. Relations and FunctionsGiven: f(x) = x2 + 2 and g(x) = 0.5x2 – 5x + 3.5Find each value. f(-3) f(x) = x2 + 2
  129. 129. Relations and FunctionsGiven: f(x) = x2 + 2 and g(x) = 0.5x2 – 5x + 3.5Find each value. f(-3) f(x) = x2 + 2 f(-3) = (-3)2 + 2
  130. 130. Relations and FunctionsGiven: f(x) = x2 + 2 and g(x) = 0.5x2 – 5x + 3.5Find each value. f(-3) f(x) = x2 + 2 f(-3) = (-3)2 + 2 f(-3) = 9 + 2
  131. 131. Relations and FunctionsGiven: f(x) = x2 + 2 and g(x) = 0.5x2 – 5x + 3.5Find each value. f(-3) f(x) = x2 + 2 f(-3) = (-3)2 + 2 f(-3) = 9 + 2 f(-3) = 11
  132. 132. Relations and FunctionsGiven: f(x) = x2 + 2 and g(x) = 0.5x2 – 5x + 3.5Find each value. f(-3) g(2.8) f(x) = x2 + 2 f(-3) = (-3)2 + 2 f(-3) = 9 + 2 f(-3) = 11

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