# Relations and Functions (Algebra 2)

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Description
Students learn about relations and functions, and the vertical line test.
Students also learn to evaluate functions
Text
• 1. Relations and Functions Analyze and graph relations. Find functional values.1) ordered pair8) function2) Cartesian Coordinate9) mapping3) plane 10) one-to-one function4) quadrant11) vertical line test5) relation12) independent variable6) domain13) dependent variable7) range 14) functional notation
• 2. Relations and FunctionsThis table shows the average lifetimeAverageMaximumand 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. Relations and FunctionsThis table shows the average lifetimeAverageMaximumand 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. Relations and FunctionsThis table shows the average lifetime AverageMaximumand maximum lifetime for some animals.AnimalLifetime Lifetime(years) (years)The data can also be represented asordered pairs. Cat12 28The ordered pairs for the data are: Cow15 30 Deer8 20 Dog12 20 Horse20 50
• 5. Relations and FunctionsThis table shows the average lifetime AverageMaximumand maximum lifetime for some animals.AnimalLifetime Lifetime(years) (years)The data can also be represented asordered pairs. Cat12 28The ordered pairs for the data are: Cow15 30 (12, 28), (15, 30), (8, 20),(12, 20), and (20, 50) Deer8 20 Dog12 20 Horse20 50
• 6. Relations and FunctionsThis table shows the average lifetime AverageMaximumand maximum lifetime for some animals.AnimalLifetime Lifetime(years) (years)The data can also be represented asordered pairs. Cat12 28The ordered pairs for the data are: Cow15 30 (12, 28), (15, 30), (8, 20),(12, 20), and (20, 50) Deer8 20The first number in each ordered pairDog12 20is the average lifetime, and the secondnumber is the maximum lifetime.Horse20 50
• 7. Relations and FunctionsThis table shows the average lifetime AverageMaximumand maximum lifetime for some animals.AnimalLifetime Lifetime(years) (years)The data can also be represented asordered pairs. Cat12 28The ordered pairs for the data are: Cow15 30 (12, 28), (15, 30), (8, 20),(12, 20), and (20, 50) Deer8 20The first number in each ordered pairDog12 20is the average lifetime, and the secondnumber is the maximum lifetime.Horse20 50(20, 50)
• 8. Relations and FunctionsThis table shows the average lifetime AverageMaximumand maximum lifetime for some animals.AnimalLifetime Lifetime(years) (years)The data can also be represented asordered pairs. Cat12 28The ordered pairs for the data are: Cow15 30 (12, 28), (15, 30), (8, 20),(12, 20), and (20, 50) Deer8 20The first number in each ordered pairDog12 20is the average lifetime, and the secondnumber is the maximum lifetime.Horse20 50(20, 50) average lifetime
• 9. Relations and FunctionsThis table shows the average lifetimeAverageMaximumand 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 pairDog 12 20is the average lifetime, and the secondnumber is the maximum lifetime.Horse 20 50(20, 50) averagemaximum lifetime lifetime
• 10. Relations and FunctionsYou can graph the ordered pairs below Animal Lifetimeson a coordinate system with two axes.y 60 50Maximum Lifetime 40 30 20 10 0 x0 5 10 15 20 25 30Average Lifetime
• 11. Relations and FunctionsYou can graph the ordered pairs below Animal Lifetimeson a coordinate system with two axes.y(12, 28),60 50Maximum Lifetime 40 30 20 10 0 x0 5 10 15 20 25 30Average Lifetime
• 12. Relations and FunctionsYou can graph the ordered pairs belowAnimal Lifetimeson a coordinate system with two axes. y(12, 28), (15, 30), 6050 Maximum Lifetime403020100 x 0 5 10 15 20 25 30 Average Lifetime
• 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 50Maximum Lifetime 40 30 20 10 0 x0 5 10 15 20 25 30Average Lifetime
• 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), 50Maximum Lifetime 40 30 20 10 0 x0 5 10 15 20 25 30Average Lifetime
• 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)50Maximum Lifetime 40 30 20 10 0 x0 5 10 15 20 25 30Average Lifetime
• 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) 50Maximum Lifetime 40Remember, each point in the coordinateplane can be named by exactly one30ordered pair and that every ordered pairnames exactly one point in the coordinate20plane. 10 0 x0 5 10 15 20 25 30Average Lifetime
• 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) 50Maximum Lifetime 40Remember, each point in the coordinateplane can be named by exactly one30ordered pair and that every ordered pairnames exactly one point in the coordinate20plane. 10The graph of this data (animal lifetimes)0 xlies in only one part of the Cartesian0 5 10 15 20 25 30coordinate plane – the part with allAverage Lifetimepositive numbers.
• 18. Relations and FunctionsThe Cartesian coordinate system is composed of the x-axis (horizontal), -505
• 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. 5Origin(0, 0) 0 -50 5 -5
• 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. 5Quadrant IIOriginQuadrant I ( --, + ) ( +, (0, )0) + 0 -505Quadrant IIIQuadrant IV ( --, -- )( +, -- ) -5
• 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. 5Quadrant IIOriginQuadrant I ( --, + ) ( +, (0, )0) + 0 -505Quadrant IIIQuadrant IV ( --, -- )( +, -- ) -5 The points on the two axes do not lie in any quadrant.
• 22. Relations and FunctionsIn general, any ordered pair in the coordinate plane can be written in the form (x, y)
• 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. 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. 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. 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. Relations and FunctionsA function is a special type of relation in which each element of the domain is pairedwith ___________ element in the range.
• 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. 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. 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. 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 -302
• 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 10 22 4
• 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 10 22 4
• 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 10 22 4
• 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 10 22 4
• 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DomainRange -3 10 22 4one-to-one function
• 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. 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DomainRange -15134
• 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DomainRange -15134
• 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DomainRange -15134
• 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DomainRange -15134
• 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DomainRange -15134 function,not one-to-one
• 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. 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 56 -3 01 1
• 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 56 -3 01 1
• 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 56 -3 01 1
• 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 56 -3 01 1not a function
• 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 56 -3 01 1not a function
• 49. Relations and FunctionsState the domain and range of the relation shownyin the graph. Is the relation a function? (-4,3) (2,3)x(-1,-2)(3,-3) (0,-4)
• 50. Relations and FunctionsState the domain and range of the relation shownyin 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. Relations and FunctionsState the domain and range of the relation shownyin 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. Relations and FunctionsState the domain and range of the relation shownyin 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. Relations and FunctionsState the domain and range of the relation shownyin 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. Relations and FunctionsState the domain and range of the relation shownyin 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. Relations and FunctionsState the domain and range of the relation shownyin 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. Relations and FunctionsYou can use the vertical line test to determine whether a relation is a function.
• 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 agraph in more than one point, the graph represents a function.yx
• 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 agraph in more than one point, the graph represents a function.yx
• 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 agraph in more than one point, the graph represents a function.yx
• 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 agraph in more than one point, the graph represents a function.yx
• 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 agraph in more than one point, the graph represents a function.yx
• 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 agraph in more than one point, the graph represents a function.yx
• 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 agraph in more than one point, the graph represents a function.yx
• 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 agraph in more than one point, the graph represents a function.yx
• 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 aIf some vertical line intercepts agraph in more than one point,graph in two or more points, the the graph represents a function.graph does not represent a function.y yx x
• 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 aIf some vertical line intercepts agraph in more than one point,graph in two or more points, the the graph represents a function.graph does not represent a function.y yx x
• 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 aIf some vertical line intercepts agraph in more than one point,graph in two or more points, the the graph represents a function.graph does not represent a function.y yx x
• 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 aIf some vertical line intercepts agraph in more than one point,graph in two or more points, the the graph represents a function.graph does not represent a function.y yx x
• 69. Relations and FunctionsThe table shows the population of Indiana over the last severalPopulationYeardecades.(millions)19503.919604.719705.219805.519905.520006.1
• 70. Relations and FunctionsThe table shows the population of Indiana over the last severalPopulationYeardecades.(millions)19503.9We can graph this data to determine 19604.7if it represents a function.19705.219805.519905.520006.1
• 71. Relations and FunctionsThe table shows the population of Indiana over the last several Population Yeardecades. (millions) 19503.9We can graph this data to determine19604.7if it represents a function. 19705.2 Population of Indiana 8 19805.5 7 6 19905.5 Population 5(millions) 20006.1 4 3 2 1 0 ‘50 ‘60 ‘70 ‘80 ‘90 ‘00 70Year
• 72. Relations and FunctionsThe table shows the population of Indiana over the last severalPopulationYeardecades.(millions)19503.9We can graph this data to determine 19604.7if it represents a function.19705.2 Population of Indiana 819805.5 7 619905.5 Use the vertical Population 5(millions) line test. 20006.1 4 3 2 1 0 ‘50 ‘60 ‘70 ‘80 ‘90 ‘00 70Year
• 73. Relations and FunctionsThe table shows the population of Indiana over the last severalPopulationYeardecades.(millions)19503.9We can graph this data to determine 19604.7if it represents a function.19705.2 Population of Indiana 819805.5 7 619905.5 Use the vertical Population 5(millions) line test. 20006.1 4 3 2 1 0 ‘50 ‘60 ‘70 ‘80 ‘90 ‘00 70Year
• 74. Relations and FunctionsThe table shows the population of Indiana over the last severalPopulationYeardecades.(millions)19503.9We can graph this data to determine 19604.7if it represents a function.19705.2 Population of Indiana 819805.5 7 619905.5 Use the vertical Population 5(millions) line test. 20006.1 4 3 2 1 0 ‘50 ‘60 ‘70 ‘80 ‘90 ‘00 70Year
• 75. Relations and FunctionsThe table shows the population of Indiana over the last severalPopulationYeardecades.(millions)19503.9We can graph this data to determine 19604.7if it represents a function.19705.2 Population of Indiana 819805.5 7 619905.5 Use the vertical Population 5(millions) line test. 20006.1 4 3 2 1 0 ‘50 ‘60 ‘70 ‘80 ‘90 ‘00 70Year
• 76. Relations and FunctionsThe table shows the population of Indiana over the last severalPopulationYeardecades.(millions)19503.9We can graph this data to determine 19604.7if it represents a function.19705.2 Population of Indiana 819805.5 7 619905.5 Use the vertical Population 5(millions) line test. 20006.1 4 3 2 1 0 ‘50 ‘60 ‘70 ‘80 ‘90 ‘00 70Year
• 77. Relations and FunctionsThe table shows the population of Indiana over the last severalPopulationYeardecades.(millions)19503.9We can graph this data to determine 19604.7if it represents a function.19705.2 Population of Indiana 819805.5 7 619905.5 Use the vertical Population 5(millions) line test. 20006.1 4 3 2 1 0 ‘50 ‘60 ‘70 ‘80 ‘90 ‘00 70Year
• 78. Relations and FunctionsThe table shows the population of Indiana over the last severalPopulationYeardecades.(millions)19503.9We can graph this data to determine 19604.7if it represents a function.19705.2 Population of Indiana 819805.5 7 619905.5 Use the vertical Population 5(millions) line test. 20006.1 4 3 2 1 0 ‘50 ‘60 ‘70 ‘80 ‘90 ‘00 70Year
• 79. Relations and FunctionsThe table shows the population of Indiana over the last severalPopulationYeardecades.(millions)19503.9We can graph this data to determine 19604.7if it represents a function.19705.2 Population of Indiana 819805.5 7 619905.5 Use the vertical Population 5(millions) line test. 20006.1 4 3 2 1 0 ‘50 ‘60 ‘70 ‘80 ‘90 ‘00 70Year
• 80. Relations and FunctionsThe table shows the population of Indiana over the last severalPopulationYeardecades.(millions)19503.9We can graph this data to determine 19604.7if it represents a function.19705.2 Population of Indiana 819805.5 7 619905.5 Use the vertical Population 5(millions) line test. 20006.1 4 3 2 1 0 ‘50 ‘60 ‘70 ‘80 ‘90 ‘00 70Year
• 81. Relations and FunctionsThe table shows the population of Indiana over the last several PopulationYeardecades. (millions)1950 3.9We can graph this data to determine 1960 4.7if it represents a function.1970 5.2 Population of Indiana 81980 5.5 7 61990 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 70Year
• 82. Relations and FunctionsThe table shows the population of Indiana over the last several PopulationYeardecades. (millions)1950 3.9We can graph this data to determine 1960 4.7if it represents a function.1970 5.2 Population of Indiana 81980 5.5 7 61990 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 0Therefore, this relation is a function! ‘50 ‘60 ‘70 ‘80 ‘90 ‘00 70Year
• 83. Relations and FunctionsGraph the relation y  2 x  1
• 84. Relations and FunctionsGraph the relation y  2 x  11) Make a table of values.
• 85. Relations and FunctionsGraph the relation y  2 x  11) Make a table of values.x y -1-10 11 32 5
• 86. Relations and FunctionsGraph the relation y  2 x  1 2) Graph the ordered pairs.1) Make a table of values.x y -1-10 11 32 5
• 87. Relations and FunctionsGraph the relation y  2 x  1 2) Graph the ordered pairs.1) Make a table of values. y76x y 54 -1-1320 110x1 3-12 5 -2-3-5 -4 -3 -2 -1 1 2 3 4 5 0
• 88. Relations and FunctionsGraph the relation y  2 x  1 2) Graph the ordered pairs.1) Make a table of values. y76x y 54 -1-1320 110x1 3-12 5 -2-3-5 -4 -3 -2 -1 1 2 3 4 5 03) Find the domain and range.
• 89. Relations and FunctionsGraph the relation y  2 x  1 2) Graph the ordered pairs.1) Make a table of values. y76x y 54 -1-1320 110x1 3-12 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. Relations and FunctionsGraph the relation y  2 x  1 2) Graph the ordered pairs.1) Make a table of values. y76x y 54 -1-1320 110x1 3-12 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. Relations and FunctionsGraph the relation y  2 x  1 2) Graph the ordered pairs.1) Make a table of values. y76x y 54 -1-1320 110x1 3-12 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. Relations and FunctionsGraph the relation y  2 x  1 2) Graph the ordered pairs.1) Make a table of values. y76x y 54 -1-1320 110x1 3-12 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. Relations and FunctionsGraph the relation y  2 x  1 2) Graph the ordered pairs.1) Make a table of values. y76x y 54 -1-1320 110x1 3-12 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. Relations and FunctionsGraph the relation y  2 x  1 2) Graph the ordered pairs.1) Make a table of values. y76x y 54 -1-1320 110x1 3-12 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. Relations and FunctionsGraph the relation y  2 x  1 2) Graph the ordered pairs.1) Make a table of values. y76x y 54 -1-1320 110x1 3-12 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. Relations and FunctionsGraph the relation y  2 x  1 2) Graph the ordered pairs.1) Make a table of values. y76x y 54 -1-1320 110x1 3-12 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. Relations and FunctionsGraph the relation y  2 x  1 2) Graph the ordered pairs.1) Make a table of values. y76x y 54 -1-1320 110x1 3-12 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. Relations and FunctionsGraph the relation x  y 2  2
• 99. Relations and FunctionsGraph the relation x  y 2  21) Make a table of values.
• 100. Relations and FunctionsGraph the relation x  y 2  21) Make a table of values.x y2-2 -1-1 -2 0 -1 12 2
• 101. Relations and FunctionsGraph the relation x  y 2  2 2) Graph the ordered pairs.1) Make a table of values.x y2-2 -1-1 -2 0 -1 12 2
• 102. Relations and FunctionsGraph the relation x  y 2  2 2) Graph the ordered pairs. y1) Make a table of values.765x y42-2 32 -1-1 10x -2 0 -1-2 -1 1 -3-5 -4 -3 -2 -1 1 2 3 4 5 02 2
• 103. Relations and FunctionsGraph the relation x  y 2  2 2) Graph the ordered pairs. y1) Make a table of values.765x y42-2 32 -1-1 10x -2 0 -1-2 -1 1 -3-5 -4 -3 -2 -1 1 2 3 4 5 02 23) Find the domain and range.
• 104. Relations and FunctionsGraph the relation x  y 2  22) Graph the ordered pairs.y1) Make a table of values. 7 6 5x y 42 -2 3 2 -1 -1 1 0x -2 0-1 -2 -1 1-3 -5 -4 -3 -2 -1 1 2 3 4 502 23) Find the domain and range. Domain is all real numbers, greater than or equal to -2.
• 105. Relations and FunctionsGraph the relation x  y 2  22) Graph the ordered pairs.y1) Make a table of values. 7 6 5x y 42 -2 3 2 -1 -1 1 0x -2 0-1 -2 -1 1-3 -5 -4 -3 -2 -1 1 2 3 4 502 23) Find the domain and range. Domain is all real numbers, greater than or equal to -2. Range is all real numbers.
• 106. Relations and FunctionsGraph the relation x  y 2  22) Graph the ordered pairs.y1) Make a table of values. 7 6 5x y 42 -2 3 2 -1 -1 1 0x -2 0-1 -2 -1 1-3 -5 -4 -3 -2 -1 1 2 3 4 502 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. Relations and FunctionsGraph the relation x  y 2  22) Graph the ordered pairs.y1) Make a table of values. 7 6 5x y 42 -2 3 2 -1 -1 1 0x -2 0-1 -2 -1 1-3 -5 -4 -3 -2 -1 1 2 3 4 502 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. Relations and FunctionsGraph the relation x  y 2  22) Graph the ordered pairs.y1) Make a table of values. 7 6 5x y 42 -2 3 2 -1 -1 1 0x -2 0-1 -2 -1 1-3 -5 -4 -3 -2 -1 1 2 3 4 502 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. Relations and FunctionsGraph the relation x  y 2  22) Graph the ordered pairs.y1) Make a table of values. 7 6 5x y 42 -2 3 2 -1 -1 1 0x -2 0-1 -2 -1 1-3 -5 -4 -3 -2 -1 1 2 3 4 502 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. Relations and FunctionsGraph the relation x  y 2  22) Graph the ordered pairs.y1) Make a table of values. 7 6 5x y 42 -2 3 2 -1 -1 1 0x -2 0-1 -2 -1 1-3 -5 -4 -3 -2 -1 1 2 3 4 502 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 – 2DOES NOT represent a function.
• 111. Relations and FunctionsWhen an equation represents a function, the variable (usually x) whose values makeup the domain is called the independent variable.
• 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. 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. 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. 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 “fof x”
• 116. Relations and FunctionsWhen an equation represents a function, the variable (usuallyx) 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 “fof x”The f is just the name of the function. It is NOT a variable that is multiplied by x.
• 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. 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. 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 asf(4) and is read“f of 4.”The value f(4) is found by substituting 4 for each x in the equation.
• 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 asf(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. 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 asf(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 Thenf(4) = 2(4) + 1
• 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 asf(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 Thenf(4) = 2(4) + 1f(4) = 8 + 1
• 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 asf(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 Thenf(4) = 2(4) + 1f(4) = 8 + 1f(4) = 9
• 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 asf(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 Thenf(4) = 2(4) + 1f(4) = 8 + 1f(4) = 9NOTE: Letters other than f can be used to represent a function.
• 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 asf(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 Thenf(4) = 2(4) + 1f(4) = 8 + 1f(4) = 9NOTE: Letters other than f can be used to represent a function.EXAMPLE: g(x) = 2x + 1
• 126. Relations and FunctionsGiven:f(x) = x2 + 2 and g(x) = 0.5x2 – 5x + 3.5Find each value.
• 127. Relations and FunctionsGiven:f(x) = x2 + 2 and g(x) = 0.5x2 – 5x + 3.5Find each value. f(-3)
• 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. Relations and FunctionsGiven:f(x) = x2 + 2 and g(x) = 0.5x2 – 5x + 3.5Find each value. f(-3)f(x) = x2 + 2f(-3) = (-3)2 + 2
• 130. Relations and FunctionsGiven:f(x) = x2 + 2 and g(x) = 0.5x2 – 5x + 3.5Find each value. f(-3)f(x) = x2 + 2f(-3) = (-3)2 + 2f(-3) = 9 + 2
• 131. Relations and FunctionsGiven:f(x) = x2 + 2 and g(x) = 0.5x2 – 5x + 3.5Find each value. f(-3)f(x) = x2 + 2f(-3) = (-3)2 + 2f(-3) = 9 + 2f(-3) = 11
• 132. Relations and FunctionsGiven:f(x) = x2 + 2 andg(x) = 0.5x2 – 5x + 3.5Find each value. f(-3)g(2.8)f(x) = x2 + 2f(-3) = (-3)2 + 2f(-3) = 9 + 2f(-3) = 11
• 133. Relations and FunctionsGiven:f(x) = x2 + 2 andg(x) = 0.5x2 – 5x + 3.5Find each value. f(-3)g(2.8)f(x) = x2 + 2 g(x) = 0.5x2 – 5x + 3.5f(-3) = (-3)2 + 2f(-3) = 9 + 2f(-3) = 11
• 134. Relations and FunctionsGiven:f(x) = x2 + 2 andg(x) = 0.5x2 – 5x + 3.5Find each value. f(-3)g(2.8)f(x) = x2 + 2 g(x) = 0.5x2 – 5x + 3.5f(-3) = (-3)2 + 2 g(2.8) = 0.5(2.8)2 – 5(2.8) + 3.5f(-3) = 9 + 2f(-3) = 11
• 135. Relations and FunctionsGiven:f(x) = x2 + 2 andg(x) = 0.5x2 – 5x + 3.5Find each value. f(-3)g(2.8)f(x) = x2 + 2 g(x) = 0.5x2 – 5x + 3.5f(-3) = (-3)2 + 2 g(2.8) = 0.5(2.8)2 – 5(2.8) + 3.5f(-3) = 9 + 2 g(2.8) = 3.92 – 14 + 3.5f(-3) = 11
• 136. Relations and FunctionsGiven:f(x) = x2 + 2 andg(x) = 0.5x2 – 5x + 3.5Find each value. f(-3)g(2.8)f(x) = x2 + 2 g(x) = 0.5x2 – 5x + 3.5f(-3) = (-3)2 + 2 g(2.8) = 0.5(2.8)2 – 5(2.8) + 3.5f(-3) = 9 + 2 g(2.8) = 3.92 – 14 + 3.5f(-3) = 11g(2.8) = – 6.58
• 137. Relations and FunctionsGiven:f(x) = x2 + 2Find the value.
• 138. Relations and FunctionsGiven:f(x) = x2 + 2Find the value.f(3z)
• 139. Relations and FunctionsGiven:f(x) = x2 + 2Find the value.f(3z) f(x) = x2 + 2
• 140. Relations and FunctionsGiven:f(x) = x2 + 2Find the value.f(3z) f(x) = x2 + 2 f( 3z ) = (3z) 2 + 2
• 141. Relations and FunctionsGiven:f(x) = x2 + 2Find the value.f(3z) f(x) = x2 + 2 f( 3z ) = (3z) 2 + 2f(3z) = 9z2 + 2
• 142. Relations and Functions
• 143. CreditsPowerPointcreated byhttp://robertfant.comUsing Glencoe’s Algebra 2 text, © 2005