9 the basic language of functions x

basic language of functions v

1. The Basic Language of Functions Math 260 Dr. Frank Ma LA H arbor College

2. The Basic Language of Functions A function is a procedure that assigns each input exactly one output.

3. The Basic Language of Functions A function is a procedure that assigns each input exactly one output. Example A. a. The inputs are driver-license numbers x, the outputs y are the names of the license-holders. N

4. The Basic Language of Functions A function is a procedure that assigns each input exactly one output. Example A. a. The inputs are driver-license numbers x, the outputs y are the names of the license-holders . This is a function because for each license number, there is exactly one owner of that license. N

5. Example A. a. The inputs are driver-license numbers x, the outputs y are the names of the license-holders. This is a function because for each license number, there is exactly one owner of that license. 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 by default we let x represent the input and y represent the output. N

6. Example A. a. The inputs are driver-license numbers x, the outputs y are the names of the license-holders. This is a function because for each license number, there is exactly one owner of that license. The Basic Language of Functions b. The inputs are driver-license numbers x, the outputs y are the names of all the siblings of the license-holders. A function is a procedure that assigns each input exactly one output. In mathematics, usually we name functions as f, g, h…and by default we let x represent the input and y represent the output.

7. Example A. a. The inputs are driver-license numbers x, the outputs y are the names of the license-holders. This is a function because for each license number, there is exactly one owner of that license. The Basic Language of Functions b. The inputs are driver-license numbers x, the outputs y are the names of all the siblings of the license-holders. This is not a function because for some inputs of license numbers, there might be many outputs of names of the siblings of the license holders. A function is a procedure that assigns each input exactly one output. In mathematics, usually we name functions as f, g, h…and by default we let x represent the input and y represent the output.

8. Given a function, the set D of all the inputs is called the domain of the function. The Basic Language of Functions

9. Given a function, the set D of all the inputs is called the domain of the function. The domain D of the license-number-to-name function in example A are all valid driver license numbers. The Basic Language of Functions N

10. Given a function, the set D of all the inputs is called the domain of the function. The domain D of the license-number-to-name function in example A are all valid driver license numbers. The set R of all the outputs is called the range and the range R of the license-number-to-name function are all the names of drivers with a valid license. The Basic Language of Functions N

11. Given a function, the set D of all the inputs is called the domain of the function. The domain D of the license-number-to-name function in example A are all valid driver license numbers. The set R of all the outputs is called the range and the range R of the license-number-to-name function are all the names of drivers with a valid license. A function may be given by a chart where the input and outputs are listed explicitly. The Basic Language of Functions x y –1 4 2 3 5 –3 6 4 7 2

12. Given a function, the set D of all the inputs is called the domain of the function. The domain D of the license-number-to-name function in example A are all valid driver license numbers. The set R of all the outputs is called the range and the range R of the license-number-to-name function are all the names of drivers with a valid license. A function may be given by a chart where the input and outputs are listed explicitly. From the 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

13. Given a function, the set D of all the inputs is called the domain of the function. The domain D of the license-number-to-name function in example A are all valid driver license numbers. The set R of all the outputs is called the range and the range R of the license-number-to-name function are all the names of drivers with a valid license. A function may be given by a chart where the input and outputs are listed explicitly. From the table we see that 3 is the output for the input 2 . The domain D = { –1, 2, 5, 6, 7 } and 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

14. Given a function, the set D of all the inputs is called the domain of the function. The domain D of the license-number-to-name function in example A are all valid driver license numbers. The set R of all the outputs is called the range and the range R of the license-number-to-name function are all the names of drivers with a valid license. A function may be given by a chart where the input and outputs are listed explicitly. From the table we see that 3 is the output for the input 2 . The domain D = { –1, 2, 5, 6, 7 } and 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

15. Functions may be given graphically: The Basic Language of Functions

16. Functions may be given graphically: For instance, Nominal Price(1975)  $0.50 The Basic Language of Functions

17. Functions may be given graphically: Domain (Nominal Price) = {year 1918  2005} For instance, Nominal Price(1975)  $0.50 The Basic Language of Functions

18. Functions may be given graphically: Domain (Nominal Price) = {year 1918  2005} Range (Nominal Price) = {$0.20  $2.51} For instance, Nominal Price(1975)  $1.00 The Basic Language of Functions

19. The Basic Language of Functions Functions may be given graphically: Inflation Adjusted Price(1975)  $1.85

20. The Basic Language of Functions Functions may be given graphically: Inflation Adjusted Price(1975)  $1.85 Domain (Inflation Adjusted Price) = {1918  2005}

21. The Basic Language of Functions Functions may be given graphically: Domain (Inflation Adjusted Price) = {1918  2005} Range (Inflation Adjusted Price) = {$1.25  $3.50} Inflation Adjusted Price(1975)  $1.85

22. The Basic Language of Functions Most functions are given by mathematics formulas.

23. For example, f (X) = X 2 – 2X + 3 = y The Basic Language of Functions Most functions are given by mathematics formulas.

24. 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.

25. 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

26. 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

27. 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

28. 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.

29. 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) =

30. 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.

31. 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.

32. 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. Function names are used with the +, –, /, and * with the obvious interpretation. The domain of this f(x) is the set of all real numbers.

33. The Basic Language of Functions Example B. Let f(x) = –3x – 3, g(x) = –2x 2 – 3. a. Evaluate f(–2)

34. The Basic Language of Functions Example B. Let f(x) = –3x – 3, g(x) = –2x 2 – 3. a. Evaluate f(–2) f(x) = –3x – 3

35. The Basic Language of Functions Example B. Let f(x) = –3x – 3, g(x) = –2x 2 – 3. a. Evaluate f(–2) f(x) = –3x – 3 f (–2)

36. The Basic Language of Functions Example B. Let f(x) = –3x – 3, g(x) = –2x 2 – 3. a. Evaluate f(–2) f(x) = –3x – 3 f (–2) copy the input

37. The Basic Language of Functions Example B. Let f(x) = –3x – 3, g(x) = –2x 2 – 3. a. Evaluate f(–2) f(x) = –3x – 3 f (–2) copy the input then paste the input where the x is

38. The Basic Language of Functions Example B. Let f(x) = –3x – 3, g(x) = –2x 2 – 3. a. Evaluate f(–2) f(x) = –3x – 3 f (–2) = –3 (–2) – 3

39. The Basic Language of Functions Example B. Let f(x) = –3x – 3, g(x) = –2x 2 – 3x + 1. a. Evaluate f(–2). f (–2) = –3 (–2) – 3 = 3

40. The Basic Language of Functions Example B. Let f(x) = –3x – 3, g(x) = –2x 2 – 3x + 1. a. Evaluate f(–2). f (–2) = –3 (–2) – 3 = 3 b. Evaluate g(–2).

41. The Basic Language of Functions Example B. Let f(x) = –3x – 3, 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

42. The Basic Language of Functions Example B. Let f(x) = –3x – 3, 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)

43. The Basic Language of Functions Example B. Let f(x) = –3x – 3, 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

44. The Basic Language of Functions Example B. Let f(x) = –3x – 3, 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 where the x’s are

45. The Basic Language of Functions Example B. Let f(x) = –3x – 3, 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

46. The Basic Language of Functions Example B. Let f(x) = –3x – 3, 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

47. The Basic Language of Functions Example B. Let f(x) = –3x – 3, 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

48. The Basic Language of Functions Example B. Let f(x) = –3x – 3, 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).

49. The Basic Language of Functions Example B. Let f(x) = –3x – 3, 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) =

50. The Basic Language of Functions Example B. Let f(x) = –3x – 3, 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

51. The Basic Language of Functions Example B. Let f(x) = –3x – 3, 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 d. Expand and simplify g (a + b ) .

52. The Basic Language of Functions Example B. Let f(x) = –3x – 3, 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 d. Expand and simplify g (a + b) . g (a + b ) = – 2 (a + b) 2 – 3 (a + b) + 1

53. The Basic Language of Functions Example B. Let f(x) = –3x – 3, 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 d. Expand and simplify g (a + b) . g (a + b ) = – 2 (a + b) 2 – 3 (a + b) + 1 = –2a 2 – 4 ab – 2b 2 – 3a – 3b + 1

54. The Basic Language of Functions e . Find the x where f(x) = g(x). f(x) = –3x – 3, g(x ) = –2x 2 – 3x + 1

55. The Basic Language of Functions e . Find the x where f(x) = g(x). Setting f(x) = g(x ) we’ve a 2 nd degree equation: –3x – 3 = –2x 2 –3x + 1 f(x) = –3x – 3, g(x ) = –2x 2 – 3x + 1

56. The Basic Language of Functions e . Find the x where f(x) = g(x). Setting f(x) = g(x ) we’ve a 2 nd degree equation: –3x – 3 = –2x 2 –3x + 1 so 2x 2 = 4 → x = ±√2 f(x) = –3x – 3, g(x ) = –2x 2 – 3x + 1

57. The Basic Language of Functions The function f(x) = c where c is a number is called a constant function, its output is the same. e . Find the x where f(x) = g(x). Setting f(x) = g(x ) we’ve a 2 nd degree equation: –3x – 3 = –2x 2 –3x + 1 so 2x 2 = 4 → x = ±√2 f(x) = –3x – 3, g(x ) = –2x 2 – 3x + 1

58. The Basic Language of Functions The function f(x) = c where c is a number is called a constant function, its output is the same. The function P (x ) = a n x n + a n-1 x n-1 + .. + a 1 x + a , with a’s as numbers, is called a polynomial function . The 1 st degree polynomials L(x ) = ax + b and the 2 nd degree polynomials Q(x ) = ax 2 + bx + c are the most basic and important functions. e . Find the x where f(x) = g(x). Setting f(x) = g(x ) we’ve a 2 nd degree equation: –3x – 3 = –2x 2 –3x + 1 so 2x 2 = 4 → x = ±√2 f(x) = –3x – 3, g(x ) = –2x 2 – 3x + 1

59. The Basic Language of Functions The function f(x) = c where c is a number is called a constant function, its output is the same. The function P (x ) = a n x n + a n-1 x n-1 + .. + a 1 x + a , with a’s as numbers, is called a polynomial function . The 1 st degree polynomials L(x ) = ax + b and the 2 nd degree polynomials Q(x ) = ax 2 + bx + c are the most basic and important functions. A rational or fractional function is a function of the form R (x) = N (x)/D(x), where N and D are polynomials . e . Find the x where f(x) = g(x). Setting f(x) = g(x ) we’ve a 2 nd degree equation: –3x – 3 = –2x 2 –3x + 1 so 2x 2 = 4 → x = ±√2 f(x) = –3x – 3, g(x ) = –2x 2 – 3x + 1

60. There are two main things to consider when determining the domains of functions of real numbers. The Basic Language of Functions

61. 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

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. The Basic Language of Functions

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) b. f (X) =  2x + 6 The Basic Language of Functions

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 b. f (X) =  2x + 6 The Basic Language of Functions

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 b. f (X) =  2x + 6 The Basic Language of Functions

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 The Basic Language of Functions

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. The Basic Language of Functions

68. 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

69. 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

70. The Basic Language of Functions Graphs of Functions

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) .

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,

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).

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). x 1 2 3 y = f(x) 1 2 3 4

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). x 1 2 3 y = f(x) 1 2 3 4 Plot the (x, y)’s and we have the graph of f(x) = x + 1, a line.

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). x 1 2 3 y = f(x) 1 2 3 4 y = x + 1 Plot the (x, y)’s and we have the graph of f(x) = x + 1, a line.

77. 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 1 2 3 y = f(x) 1 2 3 4 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.

78. 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 1 2 3 y = f(x) 1 2 3 4 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.

79. The Basic Language of Functions The equation x = y 2 , treating x as the input, is not a function.

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 outputs, y’s that satisfy 4 = y 2 ,

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 outputs, y’s that satisfy 4 = y 2 , namely y = 2 and y = –2.

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 outputs, y’s that satisfy 4 = y 2 , namely y = 2 and y = –2. So x = y 2 is not a function.

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 outputs, 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 1 1 4 4 y 1 -1 2 -2

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 outputs, 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 1 1 4 4 y 1 -1 2 -2 x 1 1 4 4 y 1 -1 2 -2 x = y 2

85. 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 outputs, 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, if we draw the vertical line x = 4, x 1 1 4 4 y 1 -1 2 -2 x 1 1 4 4 y 1 -1 2 -2 x = y 2 it intersects the graph at two points (4, 2) and (4, –2).

86. 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 outputs, 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, if we draw the vertical line x = 4, x 1 1 4 4 y 1 -1 2 -2 x 1 1 4 4 y 1 -1 2 -2 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.

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).

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

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 ),

90. 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.

91. 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). x 1 1 4 4 y 1 -1 2 -2 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.

92. 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. x y 2 4 2 3 4 3 1. x y 2 4 3 4 4 4 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

93. 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. x y = g(x) –1 4 2 3 5 –3 6 4 7 2 y = h(x) f(x) = –3x + 7 1. f(–1) 2. g(–1) 3.h(–1) 4. –f(3) 5. –g(3) 6. –h(3) 7. 3g(6) 8. 2f(2) 9. h(3) + h(0) 10. 2f(4) + 3g(2) 11. –f(4) + f(–4) 12. h(6)*[f(2)] 2

94. The Basic Language of Functions 1. f(x) = 1 2x – 6 2. f (x) =  2x – 6 C. Find the domain of the following functions. 5. f(x) = 1 (x – 2)(x + 6) 7. f (x) =  (x – 2)(x + 6) 3. f(x) = 1 3 – 2x 4. f (x) =  3 – 2x 6. f(x) = 1 x 2 – 1 8. f (x) =  1 – x 2 9. f (x) =  (x – 2)/(x + 6) 10. f (x) =  4x 2 – 9 For problem 7-10, draw sign charts (section 1.6) to solve for the domains.

95. The Basic Language of Functions 1. –f(3) 2. –g(3) 3. –h(3) 4. 3g(2) 5. 2f(2) 6. h(3) + h(0) 7. 2f(4) + 3g(2) 8. f(–3/2) 9. g(1/2) 13. f(3 + a) 14. g(3 + a) 15. g(3 + a) 16. f(a – b) 17. g(a – b) 18. f( ) a 1 19. f(a) 1 11. h(–3/2) 12. g(–1/2) 10. 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

96. The Basic Language of Functions 1 . where f(x) = 5. Exercise E. Given the functions f, g and h , f(x) = 3 x – 4 g(x) = –x 2 + 3x – 2 h(x) = 4 x + 1 2. where f(x) = –2. Find the x or x’s 3 . where g(x) = 0. 5 . where g(x) = f(x). 4. where g(x) = –3. 6. where h(x) = 1. 8. where h(x) = f(x). 7. where h(x) = x.

97. The Basic Language of Functions (Answers to odd problems) Exercise A. 1. It’s not a function 3. Not a function 5. Not a function Exercise B. 1. f(–1) = 10 3. h(–1) = 2 5. Not defined 7. 3g(6) = 12 9. h(3) + h(0) = 2 11. –f(4) + f(–4) = 24 1. x ≠ 3 Exercise C. 5. x ≠ 2 and x ≠ -6 3. x ≠ 3/2

98. The Basic Language of Functions 7.   x=-6 x=2 + + The domain of f(x) =  (x – 2)(x + 6) is   9.   x=-6 x=2 + + The domain of f(x) =  (x – 2)/(x + 6) is  

99. The Basic Language of Functions 1. –f(3) = 3 3. Not defined 5. 2f(2) = –2 7. 2f(4) + 3g(2) = –10 9. g(1/2) = –3/4 13. f(3 + a) = –2a – 3 15. g(3 + a) = –a 2 – 3a – 2 17. g(a – b) = –a 2 + 2ab – b 2 + 3a – 3b – 2 19. f(a) 1 = 1 11. h(–3/2) = –1/9 Exercise D. –2a+3 1 . x = 3 Exercise E. 3 . x = 1, 2 5 . x = ±√2 7. x = (–1±√17)/2