1.4 the basic language of functions

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1.4 the basic language of functions

  1. 1. The Basic Language of Functions Frank Ma © 2011
  2. 2. The Basic Language of Functions A function is a procedure that assigns each input exactly one output.
  3. 3. The Basic Language of Functions A function is a procedure that assigns each input exactly one output. In mathematics, usually we name functions as f, g, h…
  4. 4. The Basic Language of Functions A function is a procedure that assigns each input exactly one output. In mathematics, usually we name functions as f, g, h…and we let x represent the input and y represent the output y.
  5. 5. Example A. a. Input a number x, the output is (are) whole number(s) within ¾ of x. Is this a function? The Basic Language of Functions A function is a procedure that assigns each input exactly one output. In mathematics, usually we name functions as f, g, h…and we let x represent the input and y represent the output y.
  6. 6. Example A. a. Input a number x, the output is (are) whole number(s) within ¾ of x. Is this a function? No, this is not a function because if x = ½, there'll be two different outputs 0 or 1. The Basic Language of Functions A function is a procedure that assigns each input exactly one output. In mathematics, usually we name functions as f, g, h…and we let x represent the input and y represent the output y.
  7. 7. Example A. a. Input a number x, the output is (are) whole number(s) within ¾ of x. Is this a function? No, this is not a function because if x = ½, there'll be two different outputs 0 or 1. The Basic Language of Functions b. Input a number x, the output is the largest integer less than or equal to x. Is this a function? A function is a procedure that assigns each input exactly one output. In mathematics, usually we name functions as f, g, h…and we let x represent the input and y represent the output y.
  8. 8. Example A. a. Input a number x, the output is (are) whole number(s) within ¾ of x. Is this a function? No, this is not a function because if x = ½, there'll be two different outputs 0 or 1. The Basic Language of Functions b. Input a number x, the output is the largest integer less than or equal to x. Is this a function? This is a function. Its called the greatest integer function and it’s denoted as [x]. A function is a procedure that assigns each input exactly one output. In mathematics, usually we name functions as f, g, h…and we let x represent the input and y represent the output y.
  9. 9. Example A. a. Input a number x, the output is (are) whole number(s) within ¾ of x. Is this a function? No, this is not a function because if x = ½, there'll be two different outputs 0 or 1. The Basic Language of Functions b. Input a number x, the output is the largest integer less than or equal to x. Is this a function? This is a function. Its called the greatest integer function and it’s denoted as [x]. (so [3.1] = [3] = 3.). A function is a procedure that assigns each input exactly one output. In mathematics, usually we name functions as f, g, h…and we let x represent the input and y represent the output y.
  10. 10. Given a function, the set D of all the inputs is called the domain of the function, The Basic Language of Functions
  11. 11. Given a function, the set D of all the inputs is called the domain of the function, the set R of all the outputs is called the range of the function. The Basic Language of Functions
  12. 12. Given a function, the set D of all the inputs is called the domain of the function, the set R of all the outputs is called the range of the function. The domain D of [x] is the set of all real numbers. The Basic Language of Functions
  13. 13. Given a function, the set D of all the inputs is called the domain of the function, the set R of all the outputs is called the range of the function. The domain D of [x] is the set of all real numbers. The range R of [x] is the set of all integers. The Basic Language of Functions
  14. 14. Given a function, the set D of all the inputs is called the domain of the function, the set R of all the outputs is called the range of the function. The domain D of [x] is the set of all real numbers. The range R of [x] is the set of all integers. There are many ways to define functions. The Basic Language of Functions
  15. 15. Given a function, the set D of all the inputs is called the domain of the function, the set R of all the outputs is called the range of the function. The domain D of [x] is the set of all real numbers. The range R of [x] is the set of all integers. There are many ways to define functions. Functions may be defined by written instructions such as [x] above. The Basic Language of Functions
  16. 16. Given a function, the set D of all the inputs is called the domain of the function, the set R of all the outputs is called the range of the function. The domain D of [x] is the set of all real numbers. The range R of [x] is the set of all integers. There are many ways to define functions. Functions may be defined by written instructions such as [x] above. Functions may be given a table as shown. The Basic Language of Functions x y – 1 4 2 3 5 – 3 6 4 7 2
  17. 17. Given a function, the set D of all the inputs is called the domain of the function, the set R of all the outputs is called the range of the function. The domain D of [x] is the set of all real numbers. The range R of [x] is the set of all integers. There are many ways to define functions. Functions may be defined by written instructions such as [x] above. Functions may be given a table as shown. With this table we see that 3 is the output for the input 2 , The Basic Language of Functions x y – 1 4 2 3 5 – 3 6 4 7 2
  18. 18. Given a function, the set D of all the inputs is called the domain of the function, the set R of all the outputs is called the range of the function. The domain D of [x] is the set of all real numbers. The range R of [x] is the set of all integers. There are many ways to define functions. Functions may be defined by written instructions such as [x] above. Functions may be given a table as shown. With this table we see that 3 is the output for the input 2 , and the domain D = { –1, 2, 5, 6, 7 }, The Basic Language of Functions x y – 1 4 2 3 5 – 3 6 4 7 2
  19. 19. Given a function, the set D of all the inputs is called the domain of the function, the set R of all the outputs is called the range of the function. The domain D of [x] is the set of all real numbers. The range R of [x] is the set of all integers. There are many ways to define functions. Functions may be defined by written instructions such as [x] above. Functions may be given a table as shown. With this table we see that 3 is the output for the input 2 , and the domain D = { –1, 2, 5, 6, 7 }, the range is R = {4, 3, –3, 2}. The Basic Language of Functions x y – 1 4 2 3 5 – 3 6 4 7 2
  20. 20. Given a function, the set D of all the inputs is called the domain of the function, the set R of all the outputs is called the range of the function. The domain D of [x] is the set of all real numbers. The range R of [x] is the set of all integers. There are many ways to define functions. Functions may be defined by written instructions such as [x] above. Functions may be given a table as shown. With this table we see that 3 is the output for the input 2 , and the domain D = { –1, 2, 5, 6, 7 }, the range is R = {4, 3, –3, 2}. Note that we may have the same output 4 for two different inputs –1 and 6 . The Basic Language of Functions x y – 1 4 2 3 5 – 3 6 4 7 2
  21. 21. Functions may be given graphically: The Basic Language of Functions
  22. 22. Functions may be given graphically: For instance, Nominal Price(1975)  $0.50 The Basic Language of Functions
  23. 23. Functions may be given graphically: Domain = {year 1918  2005} For instance, Nominal Price(1975)  $0.50 The Basic Language of Functions
  24. 24. Functions may be given graphically: Domain = {year 1918  2005} Range (Nominal Price) = {$0.20  $2.51} For instance, Nominal Price(1975)  $1.00 The Basic Language of Functions
  25. 25. The Basic Language of Functions Functions may be given graphically: Inflation Adjusted Price(1975)  $1.85
  26. 26. The Basic Language of Functions Functions may be given graphically: Domain = {year 1918  2005} Inflation Adjusted Price(1975)  $1.85
  27. 27. The Basic Language of Functions Functions may be given graphically: Domain = {year 1918  2005} Range (Inflation Adjusted Price) = {$1.25  $3.50} Inflation Adjusted Price(1975)  $1.85
  28. 28. The Basic Language of Functions Most functions are given by mathematics formulas.
  29. 29. For example, f (X) = X 2 – 2X + 3 = y The Basic Language of Functions Most functions are given by mathematics formulas.
  30. 30. For example, f (X) = X 2 – 2X + 3 = y name of the function The Basic Language of Functions Most functions are given by mathematics formulas.
  31. 31. For example, f (X) = X 2 – 2X + 3 = y name of the function The Basic Language of Functions Most functions are given by mathematics formulas. input box
  32. 32. For example, f (X) = X 2 – 2X + 3 = y name of actual formula the function The Basic Language of Functions Most functions are given by mathematics formulas. input box
  33. 33. For example, f (X) = X 2 – 2X + 3 = y name of actual formula the function The output The Basic Language of Functions Most functions are given by mathematics formulas. input box
  34. 34. For example, f (X) = X 2 – 2X + 3 = y name of actual formula the function The output The Basic Language of Functions Most functions are given by mathematics formulas. input box The input box holds the input for the formula.
  35. 35. For example, f (X) = X 2 – 2X + 3 = y name of actual formula the function The output The Basic Language of Functions Most functions are given by mathematics formulas. input box The input box holds the input for the formula. Hence f (2) means to replace x by (2) in the formula, so f (2) = (2) 2 – 2 (2) + 3 = 3 = y.
  36. 36. For example, f (X) = X 2 – 2X + 3 = y name of actual formula the function The output The Basic Language of Functions Most functions are given by mathematics formulas. input box The input box holds the input for the formula. Hence f (2) means to replace x by (2) in the formula, so f (2) = (2) 2 – 2 (2) + 3 = 3 = y. The domain of this f(x) is the set of all real numbers.
  37. 37. For example, f (X) = X 2 – 2X + 3 = y name of actual formula the function The output The Basic Language of Functions Most functions are given by mathematics formulas. input box The input box holds the input for the formula. Hence f (2) means to replace x by (2) in the formula, so f (2) = (2) 2 – 2 (2) + 3 = 3 = y. The above function notation is used with the +, –, /, and * with the obvious interpretation. The domain of this f(x) is the set of all real numbers.
  38. 38. The Basic Language of Functions Example B. Let f(x) = –3x + 2, g(x) = –2x 2 – 3. a. Evaluate f(–2)
  39. 39. The Basic Language of Functions Example B. Let f(x) = –3x + 2, g(x) = –2x 2 – 3. a. Evaluate f(–2) f(x) = –3x + 2
  40. 40. The Basic Language of Functions Example B. Let f(x) = –3x + 2, g(x) = –2x 2 – 3. a. Evaluate f(–2) f(x) = –3x + 2 f (–2)
  41. 41. The Basic Language of Functions Example B. Let f(x) = –3x + 2, g(x) = –2x 2 – 3. a. Evaluate f(–2) f(x) = –3x + 2 f (–2) copy the input
  42. 42. The Basic Language of Functions Example B. Let f(x) = –3x + 2, g(x) = –2x 2 – 3. a. Evaluate f(–2) f(x) = –3x + 2 f (–2) copy the input then paste the input at where the x is
  43. 43. The Basic Language of Functions Example B. Let f(x) = –3x + 2, g(x) = –2x 2 – 3. a. Evaluate f(–2) f(x) = –3x + 2 f (–2) = –3 (–2) + 2
  44. 44. The Basic Language of Functions Example B. Let f(x) = –3x + 2, g(x) = –2x 2 – 3x + 1. a. Evaluate f(–2). f (–2) = –3 (–2) – 3 = 3
  45. 45. The Basic Language of Functions Example B. Let f(x) = –3x + 2, g(x) = –2x 2 – 3x + 1. a. Evaluate f(–2). f (–2) = –3 (–2) – 3 = 3 b. Evaluate g(–2).
  46. 46. The Basic Language of Functions Example B. Let f(x) = –3x + 2, g(x) = –2x 2 – 3x + 1. a. Evaluate f(–2). f (–2) = –3 (–2) – 3 = 3 b. Evaluate g(–2). g(x) = –2x 2 – 3x + 1
  47. 47. The Basic Language of Functions Example B. Let f(x) = –3x + 2, g(x) = –2x 2 – 3x + 1. a. Evaluate f(–2). f (–2) = –3 (–2) – 3 = 3 b. Evaluate g(–2). g(x) = –2x 2 – 3x + 1 g (–2)
  48. 48. The Basic Language of Functions Example B. Let f(x) = –3x + 2, g(x) = –2x 2 – 3x + 1. a. Evaluate f(–2). f (–2) = –3 (–2) – 3 = 3 b. Evaluate g(–2). g(x) = –2x 2 – 3x + 1 g (–2) copy the input
  49. 49. The Basic Language of Functions Example B. Let f(x) = –3x + 2, g(x) = –2x 2 – 3x + 1. a. Evaluate f(–2). f (–2) = –3 (–2) – 3 = 3 b. Evaluate g(–2). g(x) = –2x 2 – 3x + 1 g (–2) copy the input then paste the input at where the x’s are
  50. 50. The Basic Language of Functions Example B. Let f(x) = –3x + 2, g(x) = –2x 2 – 3x + 1. a. Evaluate f(–2). f (–2) = –3 (–2) – 3 = 3 b. Evaluate g(–2). g(x) = –2x 2 – 3x + 1 g (–2) = –2 (–2) 2 – 3 (–2) + 1 copy the input
  51. 51. The Basic Language of Functions Example B. Let f(x) = –3x + 2, g(x) = –2x 2 – 3x + 1. a. Evaluate f(–2). f (–2) = –3 (–2) – 3 = 3 b. Evaluate g(–2). g (–2) = –2 (–2) 2 – 3 (–2) + 1
  52. 52. The Basic Language of Functions Example B. Let f(x) = –3x + 2, g(x) = –2x 2 – 3x + 1. a. Evaluate f(–2). f (–2) = –3 (–2) – 3 = 3 b. Evaluate g(–2). g (–2) = –2 (–2) 2 – 3 (–2) + 1 = –8 + 6 + 1 = –1
  53. 53. The Basic Language of Functions Example B. Let f(x) = –3x + 2, g(x) = –2x 2 – 3x + 1. a. Evaluate f(–2). f (–2) = –3 (–2) – 3 = 3 b. Evaluate g(–2). g (–2) = –2 (–2) 2 – 3 (–2) + 1 = –8 + 6 + 1 = –1 c. Evaluate f(–2) – g(–2).
  54. 54. The Basic Language of Functions Example B. Let f(x) = –3x + 2, g(x) = –2x 2 – 3x + 1. a. Evaluate f(–2). f (–2) = –3 (–2) – 3 = 3 b. Evaluate g(–2). g (–2) = –2 (–2) 2 – 3 (–2) + 1 = –8 + 6 + 1 = –1 c. Evaluate f(–2) – g(–2). Using the outputs of parts a and b we’ve f(–2) – g(–2) =
  55. 55. The Basic Language of Functions Example B. Let f(x) = –3x + 2, g(x) = –2x 2 – 3x + 1. a. Evaluate f(–2). f (–2) = –3 (–2) – 3 = 3 b. Evaluate g(–2). g (–2) = –2 (–2) 2 – 3 (–2) + 1 = –8 + 6 + 1 = –1 c. Evaluate f(–2) – g(–2). Using the outputs of parts a and b we’ve f(–2) – g(–2) = 3 – (–1) = 4
  56. 56. The Basic Language of Functions Example B. Let f(x) = –3x + 2, g(x) = –2x 2 – 3x + 1. a. Evaluate f(–2). f (–2) = –3 (–2) – 3 = 3 b. Evaluate g(–2). g (–2) = –2 (–2) 2 – 3 (–2) + 1 = –8 + 6 + 1 = –1 c. Evaluate f(–2) – g(–2). Using the outputs of parts a and b we’ve f(–2) – g(–2) = 3 – (–1) = 4 The function f(x) = c where c is a number is called a constant function.
  57. 57. The Basic Language of Functions Example B. Let f(x) = –3x + 2, g(x) = –2x 2 – 3x + 1. a. Evaluate f(–2). f (–2) = –3 (–2) – 3 = 3 b. Evaluate g(–2). g (–2) = –2 (–2) 2 – 3 (–2) + 1 = –8 + 6 + 1 = –1 c. Evaluate f(–2) – g(–2). Using the outputs of parts a and b we’ve f(–2) – g(–2) = 3 – (–1) = 4 The function f(x) = c where c is a number is called a constant function. The outputs of such functions do not change.
  58. 58. There are two main things to consider when determining the domains of functions of real numbers. The Basic Language of Functions
  59. 59. There are two main things to consider when determining the domains of functions of real numbers. 1. The denominators can't be 0 The Basic Language of Functions
  60. 60. There are two main things to consider when determining the domains of functions of real numbers. 1. The denominators can't be 0 2. The radicand of square root (or any even root) can't be negative. The Basic Language of Functions
  61. 61. There are two main things to consider when determining the domains of functions of real numbers. 1. The denominators can't be 0 2. The radicand of square root (or any even root) can't be negative. Example C. Find the domain of the following functions. a. f(x) = 1/(2x + 6) b. f (X) =  2x + 6 The Basic Language of Functions
  62. 62. There are two main things to consider when determining the domains of functions of real numbers. 1. The denominators can't be 0 2. The radicand of square root (or any even root) can't be negative. Example C. Find the domain of the following functions. a. f(x) = 1/(2x + 6) The denominator can’t be 0 b. f (X) =  2x + 6 The Basic Language of Functions
  63. 63. There are two main things to consider when determining the domains of functions of real numbers. 1. The denominators can't be 0 2. The radicand of square root (or any even root) can't be negative. Example C. Find the domain of the following functions. a. f(x) = 1/(2x + 6) The denominator can’t be 0 i.e. 2x + 6 = 0  x = -3 b. f (X) =  2x + 6 The Basic Language of Functions
  64. 64. There are two main things to consider when determining the domains of functions of real numbers. 1. The denominators can't be 0 2. The radicand of square root (or any even root) can't be negative. Example C. Find the domain of the following functions. a. f(x) = 1/(2x + 6) The denominator can’t be 0 i.e. 2x + 6 = 0  x = -3 So the domain = { all numbers except x = -3} . b. f (X) =  2x + 6 The Basic Language of Functions
  65. 65. There are two main things to consider when determining the domains of functions of real numbers. 1. The denominators can't be 0 2. The radicand of square root (or any even root) can't be negative. Example C. Find the domain of the following functions. a. f(x) = 1/(2x + 6) The denominator can’t be 0 i.e. 2x + 6 = 0  x = -3 So the domain = { all numbers except x = -3} . b. f (X) =  2x + 6 We must have square root of nonnegative numbers. The Basic Language of Functions
  66. 66. There are two main things to consider when determining the domains of functions of real numbers. 1. The denominators can't be 0 2. The radicand of square root (or any even root) can't be negative. Example C. Find the domain of the following functions. a. f(x) = 1/(2x + 6) The denominator can’t be 0 i.e. 2x + 6 = 0  x = -3 So the domain = { all numbers except x = -3} . b. f (X) =  2x + 6 We must have square root of nonnegative numbers. Hence 2x + 6 > 0  x > -3 The Basic Language of Functions
  67. 67. There are two main things to consider when determining the domains of functions of real numbers. 1. The denominators can't be 0 2. The radicand of square root (or any even root) can't be negative. Example C. Find the domain of the following functions. a. f(x) = 1/(2x + 6) The denominator can’t be 0 i.e. 2x + 6 = 0  x = -3 So the domain = { all numbers except x = -3} . b. f (X) =  2x + 6 We must have square root of nonnegative numbers. Hence 2x + 6 > 0  x > -3 So the domain = { all numbers x > -3 } The Basic Language of Functions
  68. 68. The Basic Language of Functions Graphs of Functions
  69. 69. The Basic Language of Functions Graphs of Functions The plot of all points (x, y)’s that satisfy a given function y = f(x) is the graph of the function y = f(x) .
  70. 70. The Basic Language of Functions Graphs of Functions The plot of all points (x, y)’s that satisfy a given function y = f(x) is the graph of the function y = f(x) . For example, let the function f(x) = x + 1,
  71. 71. The Basic Language of Functions Graphs of Functions The plot of all points (x, y)’s that satisfy a given function y = f(x) is the graph of the function y = f(x) . For example, let the function f(x) = x + 1, set y = f(x) and make a table of few of the ordered pairs (x, y)’s that satisfy the equation y = f(x).
  72. 72. The Basic Language of Functions Graphs of Functions The plot of all points (x, y)’s that satisfy a given function y = f(x) is the graph of the function y = f(x) . For example, let the function f(x) = x + 1, set y = f(x) and make a table of few of the ordered pairs (x, y)’s that satisfy the equation y = f(x). x 0 1 2 3 y = f(x) 1 2 3 4
  73. 73. The Basic Language of Functions Graphs of Functions The plot of all points (x, y)’s that satisfy a given function y = f(x) is the graph of the function y = f(x) . For example, let the function f(x) = x + 1, set y = f(x) and make a table of few of the ordered pairs (x, y)’s that satisfy the equation y = f(x). Plot the (x, y)’s and we have the graph of f(x) = x + 1, a line. x 0 1 2 3 y = f(x) 1 2 3 4
  74. 74. The Basic Language of Functions Graphs of Functions The plot of all points (x, y)’s that satisfy a given function y = f(x) is the graph of the function y = f(x) . For example, let the function f(x) = x + 1, set y = f(x) and make a table of few of the ordered pairs (x, y)’s that satisfy the equation y = f(x). y = x + 1 Plot the (x, y)’s and we have the graph of f(x) = x + 1, a line. x 0 1 2 3 y = f(x) 1 2 3 4
  75. 75. The Basic Language of Functions Graphs of Functions The plot of all points (x, y)’s that satisfy a given function y = f(x) is the graph of the function y = f(x) . For example, let the function f(x) = x + 1, set y = f(x) and make a table of few of the ordered pairs (x, y)’s that satisfy the equation y = f(x). y = x + 1 Plot the (x, y)’s and we have the graph of f(x) = x + 1, a line. Note that the graph of a function may cross any vertical line at most at one point because for each x there is only one corresponding output y. x 0 1 2 3 y = f(x) 1 2 3 4
  76. 76. The Basic Language of Functions Graphs of Functions The plot of all points (x, y)’s that satisfy a given function y = f(x) is the graph of the function y = f(x) . For example, let the function f(x) = x + 1, set y = f(x) and make a table of few of the ordered pairs (x, y)’s that satisfy the equation y = f(x). y = x + 1 Plot the (x, y)’s and we have the graph of f(x) = x + 1, a line. Note that the graph of a function may cross any vertical line at most at one point because for each x there is only one corresponding output y. x 0 1 2 3 y = f(x) 1 2 3 4
  77. 77. The Basic Language of Functions The equation x = y 2 , treating x as the input, is not a function.
  78. 78. The Basic Language of Functions The equation x = y 2 , treating x as the input, is not a function. For instance, for the input x = 4, there are two output y’s that satisfy 4 = y 2 ,
  79. 79. The Basic Language of Functions The equation x = y 2 , treating x as the input, is not a function. For instance, for the input x = 4, there are two output y’s that satisfy 4 = y 2 , namely y = 2 and y = –2.
  80. 80. The Basic Language of Functions The equation x = y 2 , treating x as the input, is not a function. For instance, for the input x = 4, there are two output y’s that satisfy 4 = y 2 , namely y = 2 and y = –2. So x = y 2 is not a function.
  81. 81. The Basic Language of Functions The equation x = y 2 , treating x as the input, is not a function. For instance, for the input x = 4, there are two output y’s that satisfy 4 = y 2 , namely y = 2 and y = –2. So x = y 2 is not a function. Plot the graph by the table shown. x 0 1 1 4 4 y 0 1 -1 2 -2
  82. 82. The Basic Language of Functions The equation x = y 2 , treating x as the input, is not a function. For instance, for the input x = 4, there are two output y’s that satisfy 4 = y 2 , namely y = 2 and y = –2. So x = y 2 is not a function. Plot the graph by the table shown. x = y 2 x 0 1 1 4 4 y 0 1 -1 2 -2 x 0 1 1 4 4 y 0 1 -1 2 -2
  83. 83. The Basic Language of Functions The equation x = y 2 , treating x as the input, is not a function. For instance, for the input x = 4, there are two output y’s that satisfy 4 = y 2 , namely y = 2 and y = –2. So x = y 2 is not a function. Plot the graph by the table shown. In particular that if we draw the vertical line x = 4, x = y 2 it intersects the graph at two points (4, 2) and (4, –2). x 0 1 1 4 4 y 0 1 -1 2 -2 x 0 1 1 4 4 y 0 1 -1 2 -2
  84. 84. The Basic Language of Functions The equation x = y 2 , treating x as the input, is not a function. For instance, for the input x = 4, there are two output y’s that satisfy 4 = y 2 , namely y = 2 and y = –2. So x = y 2 is not a function. Plot the graph by the table shown. In particular that if we draw the vertical line x = 4, x = y 2 it intersects the graph at two points (4, 2) and (4, –2). In general if any vertical line crosses a graph at two or more points then the graph does not represent any function. x 0 1 1 4 4 y 0 1 -1 2 -2 x 0 1 1 4 4 y 0 1 -1 2 -2
  85. 85. The Basic Language of Functions Since for functions each input x has exactly one output, therefore each vertical line can only intersect it’s graph at exactly one location (e.g. y = x + 1).
  86. 86. The Basic Language of Functions Since for functions each input x has exactly one output, therefore each vertical line can only intersect it’s graph at exactly one location (e.g. y = x + 1). y = x + 1
  87. 87. The Basic Language of Functions Since for functions each input x has exactly one output, therefore each vertical line can only intersect it’s graph at exactly one location (e.g. y = x + 1). y = x + 1 However, if any vertical line intersects a graph at two or more points, i.e. there are two or more outputs y associated to one input x (eg. x = y 2 ),
  88. 88. The Basic Language of Functions Since for functions each input x has exactly one output, therefore each vertical line can only intersect it’s graph at exactly one location (e.g. y = x + 1). y = x + 1 However, if any vertical line intersects a graph at two or more points, i.e. there are two or more outputs y associated to one input x (eg. x = y 2 ), then the graph must not be the graph of a function.
  89. 89. The Basic Language of Functions Since for functions each input x has exactly one output, therefore each vertical line can only intersect it’s graph at exactly one location (e.g. y = x + 1). y = x + 1 However, if any vertical line intersects a graph at two or more points, i.e. there are two or more outputs y associated to one input x (eg. x = y 2 ), then the graph must not be the graph of a function. x 0 1 1 4 4 y 0 1 -1 2 -2
  90. 90. The Basic Language of Functions Exercise A. For problems 1 – 6, determine if the given represents a function. If it’s not a function, give a reason why it’s not. 1. 2. 3. 4. x y y 6. For any real number input x that is a rational number, the output is 0, otherwise the output is 1 5. For any input x that is a positive integer, the outputs are it’s factors. x All the (x, y)’s on the curve x y 2 4 2 3 4 3 x y 2 4 3 4 4 4
  91. 91. The Basic Language of Functions Exercise B. Given the functions f, g and h, find the outcomes of the following expressions. If it’s not defined, state so. y = h(x) f(x) = –3x + 7 7. f(–1) 8.g(–1) 9.h(–1) 10. –f(3) 11. –g(3) 12. –h(3) 13. 3g(6) 14.2f(2) 15. h(3) + h(0) 16. 2f(4) + 3g(2) 17. –f(4) + f(–4) 18. h(6)*[f(2)] 2 x y = g(x) – 1 4 2 3 5 – 3 6 4 7 2
  92. 92. The Basic Language of Functions 19. f(x) = 1 2x – 6 20. f (x) =  2x – 6 Exercise C. Find the domain of the following functions. 23. f(x) = 1 (x – 2)(x + 6) 24. f (x) =  (x – 2)(x + 6) 21. f(x) = 1 3 – 2x 22. f (x) =  3 – 2x 25. f(x) = 1 x 2 – 1 26. f (x) =  1 – x 2
  93. 93. The Basic Language of Functions 27. –f(3) 28. –g(3) 29. –h(3) 30. 3g(2) 31.2f(2) 32. h(3) + h(0) 33. 2f(4) + 3g(2) 34. f(–3/2) 35. g(1/2) 39. f(3a) 40. g(3a) 41. 3g(a) 42. g(a – b) 43. 2f(a – b) 44. f( ) a 1 45. f(a) 1 37. h(–3/2) 38. g(–1/2) 36. 1/g(2) Exercise D. Given the functions f, g and h, find the outcomes of the following expressions. If it’s not defined, state so. f(x) = –2x + 3 g(x) = –x 2 + 3x – 2 h(x) = x + 2 x – 3

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