1. 7.1 – Operations on Functions

2. OperationDefinition

3. Sum

4. OperationDefinition Sum(f + g)(x)

5. OperationDefinition Sum(f + g)(x) = f(x) + g(x)

6. OperationDefinition Sum(f + g)(x) = f(x) + g(x) Difference

7. OperationDefinition Sum(f + g)(x) = f(x) + g(x) Difference(f – g)(x) =

8. OperationDefinition Sum(f + g)(x) = f(x) + g(x) Difference(f – g)(x) = f(x) – g(x)

9. OperationDefinition Sum(f + g)(x) = f(x) + g(x) Difference(f – g)(x) = f(x) – g(x) Product

10. OperationDefinition Sum(f + g)(x) = f(x) + g(x) Difference(f – g)(x) = f(x) – g(x) Product(f · g)(x) =

11. OperationDefinition Sum(f + g)(x) = f(x) + g(x) Difference(f – g)(x) = f(x) – g(x) Product(f · g)(x) = f(x) · g(x)

12. OperationDefinition Sum(f + g)(x) = f(x) + g(x) Difference(f – g)(x) = f(x) – g(x) Product(f · g)(x) = f(x) · g(x) Quotient f (x) = g

13. OperationDefinition Sum(f + g)(x) = f(x) + g(x) Difference(f – g)(x) = f(x) – g(x) Product(f · g)(x) = f(x) · g(x) Quotient f (x) = f(x) g g(x)

14. Ex. 1 Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x)for f(x) g and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9

15. Ex. 1 Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x)for f(x) g and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9 (f + g)(x)

16. Ex. 1 Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x)for f(x) g and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9 (f + g)(x) = f(x) + g(x)

17. Ex. 1 Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x)for f(x) g and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9 (f + g)(x) = f(x) + g(x)

18. Ex. 1 Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x)for f(x) g and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9 (f + g)(x) = f(x) + g(x) = (2x – 3)

19. Ex. 1 Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x)for f(x) g and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9 (f + g)(x) = f(x) + g(x) = (2x – 3)

20. Ex. 1 Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x)for f(x) g and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9 (f + g)(x) = f(x) + g(x) = (2x – 3) + (4x + 9)

21. Ex. 1 Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x)for f(x) g and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9 (f + g)(x) = f(x) + g(x) = (2x – 3) + (4x + 9) = 6x + 6

22. Ex. 1 Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x)for f(x) g and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9 (f + g)(x) = f(x) + g(x) = (2x – 3) + (4x + 9) = 6x – 6 (f – g)(x)

23. Ex. 1 Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x)for f(x) g and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9 (f + g)(x) = f(x) + g(x) = (2x – 3) + (4x + 9) = 6x – 6 (f – g)(x) = f(x) – g(x)

24. Ex. 1 Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x)for f(x) g and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9 (f + g)(x) = f(x) + g(x) = (2x – 3) + (4x + 9) = 6x – 6 (f – g)(x) = f(x) – g(x)

25. Ex. 1 Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x)for f(x) g and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9 (f + g)(x) = f(x) + g(x) = (2x – 3) + (4x + 9) = 6x – 6 (f – g)(x) = f(x) – g(x) = (2x – 3)

26. Ex. 1 Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x)for f(x) g and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9 (f + g)(x) = f(x) + g(x) = (2x – 3) + (4x + 9) = 6x – 6 (f – g)(x) = f(x) – g(x) = (2x – 3)

27. Ex. 1 Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x)for f(x) g and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9 (f + g)(x) = f(x) + g(x) = (2x – 3) + (4x + 9) = 6x – 6 (f – g)(x) = f(x) – g(x) = (2x – 3) – (4x + 9)

28. Ex. 1 Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x)for f(x) g and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9 (f + g)(x) = f(x) + g(x) = (2x – 3) + (4x + 9) = 6x – 6 (f – g)(x) = f(x) – g(x) = (2x – 3) – (4x + 9)

29. Ex. 1 Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x)for f(x) g and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9 (f + g)(x) = f(x) + g(x) = (2x – 3) + (4x + 9) = 6x – 6 (f – g)(x) = f(x) – g(x) = (2x – 3) – (4x + 9) = 2x – 3 – 4x

30. Ex. 1 Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x)for f(x) g and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9 (f + g)(x) = f(x) + g(x) = (2x – 3) + (4x + 9) = 6x – 6 (f – g)(x) = f(x) – g(x) = (2x – 3) – (4x + 9) = 2x – 3 – 4x – 9

31. Ex. 1 Find (f + g)(x), (f – g)(x), (f · g)(x), & f (x)for f(x) g and g(x) if f(x) = 2x – 3 and g(x) = 4x + 9 (f + g)(x) = f(x) + g(x) = (2x – 3) + (4x + 9) = 6x – 6 (f – g)(x) = f(x) – g(x) = (2x – 3) – (4x + 9) = 2x – 3 – 4x – 9 = -2x – 12

32. (f · g)(x)

33. (f · g)(x) = f(x) · g(x)

35. = (2x – 3)

36. (f · g)(x) = f(x) · g(x) = (2x – 3)

37. (f · g)(x) = f(x) · g(x) = (2x – 3)(4x + 9)

38. (f · g)(x) = f(x) · g(x) = (2x – 3)(4x + 9) = 8x 2 + 18x – 12x – 27

39. (f · g)(x) = f(x) · g(x) = (2x – 3)(4x + 9) = 8x 2 + 18x – 12x – 27 = 8x 2 + 6x – 27

40. (f · g)(x) = f(x) · g(x) = (2x – 3)(4x + 9) = 8x 2 + 18x – 12x – 27 = 8x 2 + 6x – 27 f (x) g

41. (f · g)(x) = f(x) · g(x) = (2x – 3)(4x + 9) = 8x 2 + 18x – 12x – 27 = 8x 2 + 6x – 27 f (x) = f(x) g g(x)

42. (f · g)(x) = f(x) · g(x) = (2x – 3)(4x + 9) = 8x 2 + 18x – 12x – 27 = 8x 2 + 6x – 27 f (x) = f(x) g g(x) = 2x – 3 4x + 9

43. (f · g)(x) = f(x) · g(x) = (2x – 3)(4x + 9) = 8x 2 + 18x – 12x – 27 = 8x 2 + 6x – 27 f (x) = f(x) g g(x) = 2x – 3 4x + 9 *Factor & Simplify if possible!

44. Composite Function

45. - taking the function

46. Composite Function - taking the function of a function

47. Composite Function - taking the function of a function [f °g(x)]

48. Composite Function - taking the function of a function [f °g(x)] = f[g(x)]

49. Composite Function - taking the function of a function [f °g(x)] = f[g(x)] Ex. 2 Find [f °g(x)] and [g°f(x)] for the functions f(x) = x + 3 and g(x) = x 2 + x – 1.

50. Composite Function - taking the function of a function [f °g(x)] = f[g(x)] Ex. 2 Find [f °g(x)] and [g°f(x)] for the functions f(x) = x + 3 and g(x) = x 2 + x – 1. [f °g(x)] = f[g(x)]

51. Composite Function - taking the function of a function [f °g(x)] = f[g(x)] Ex. 2 Find [f °g(x)] and [g°f(x)] for the functions f(x) = x + 3 and g(x) = x 2 + x – 1. [f °g(x)] = f[g(x)]

52. Composite Function - taking the function of a function [f °g(x)] = f[g(x)] Ex. 2 Find [f °g(x)] and [g°f(x)] for the functions f(x) = x + 3 and g(x) = x 2 + x – 1. [f °g(x)] = f[g(x)] = f[x 2 + x – 1]

53. Composite Function - taking the function of a function [f °g(x)] = f[g(x)] Ex. 2 Find [f °g(x)] and [g°f(x)] for the functions f(x) = x + 3 and g(x) = x 2 + x – 1. [f °g(x)] = f[g(x)] = f[x 2 + x – 1]

54. Composite Function - taking the function of a function [f °g(x)] = f[g(x)] Ex. 2 Find [f °g(x)] and [g°f(x)] for the functions f(x) = x + 3 and g(x) = x 2 + x – 1. [f °g(x)] = f[g(x)] = f[x 2 + x – 1]

55. Composite Function - taking the function of a function [f °g(x)] = f[g(x)] Ex. 2 Find [f °g(x)] and [g°f(x)] for the functions f(x) = x + 3 and g(x) = x 2 + x – 1. [f °g(x)] = f[g(x)] = f(x 2 + x – 1) = (x 2 + x – 1) + 3

56. Composite Function - taking the function of a function [f °g(x)] = f[g(x)] Ex. 2 Find [f °g(x)] and [g°f(x)] for the functions f(x) = x + 3 and g(x) = x 2 + x – 1. [f °g(x)] = f[g(x)] = f(x 2 + x – 1) = (x 2 + x – 1) + 3 = x 2 + x + 2

57. f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)]

58. f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)]

59. f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)]

60. f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)]

61. f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3)

62. f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3)

63. f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3)

64. f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3) = (x + 3) 2

65. f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3) = (x + 3) 2

66. f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3) = (x + 3) 2 + (x + 3)

67. f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3) = (x + 3) 2 + (x + 3)

68. f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3) = (x + 3) 2 + (x + 3) – 1

69. f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3) = (x + 3) 2 + (x + 3) – 1 = (x + 3)(x + 3) + (x + 3) – 1

70. f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3) = (x + 3) 2 + (x + 3) – 1 = (x + 3)(x + 3) + (x + 3) – 1 = x 2 + 6x + 9 + x + 3 – 1

71. f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3) = (x + 3) 2 + (x + 3) – 1 = (x + 3)(x + 3) + (x + 3) – 1 = x 2 + 6x + 9 + x + 3 – 1 = x 2 + 7x + 11

72. f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3) = (x + 3) 2 + (x + 3) – 1 = (x + 3)(x + 3) + (x + 3) – 1 = x 2 + 6x + 9 + x + 3 – 1 = x 2 + 7x + 11 Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)].

73. f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3) = (x + 3) 2 + (x + 3) – 1 = (x + 3)(x + 3) + (x + 3) – 1 = x 2 + 6x + 9 + x + 3 – 1 = x 2 + 7x + 11 Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)]. g[f(5)] =

74. f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3) = (x + 3) 2 + (x + 3) – 1 = (x + 3)(x + 3) + (x + 3) – 1 = x 2 + 6x + 9 + x + 3 – 1 = x 2 + 7x + 11 Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)]. g[f(5)] =

75. f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3) = (x + 3) 2 + (x + 3) – 1 = (x + 3)(x + 3) + (x + 3) – 1 = x 2 + 6x + 9 + x + 3 – 1 = x 2 + 7x + 11 Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)]. g[f(5)] =

76. f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3) = (x + 3) 2 + (x + 3) – 1 = (x + 3)(x + 3) + (x + 3) – 1 = x 2 + 6x + 9 + x + 3 – 1 = x 2 + 7x + 11 Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)]. g[f(5)] = g[4(5)]

77. f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3) = (x + 3) 2 + (x + 3) – 1 = (x + 3)(x + 3) + (x + 3) – 1 = x 2 + 6x + 9 + x + 3 – 1 = x 2 + 7x + 11 Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)]. g[f(5)] = g[4(5)] = g(20)

78. f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3) = (x + 3) 2 + (x + 3) – 1 = (x + 3)(x + 3) + (x + 3) – 1 = x 2 + 6x + 9 + x + 3 – 1 = x 2 + 7x + 11 Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)]. g[f(5)] = g[4(5)] = g(20)

79. f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3) = (x + 3) 2 + (x + 3) – 1 = (x + 3)(x + 3) + (x + 3) – 1 = x 2 + 6x + 9 + x + 3 – 1 = x 2 + 7x + 11 Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)]. g[f(5)] = g[4(5)] = g(20) = 2(20) – 1

80. f(x) = x + 3 and g(x) = x 2 + x – 1 [g°f(x)] = g[f(x)] = g(x + 3) = (x + 3) 2 + (x + 3) – 1 = (x + 3)(x + 3) + (x + 3) – 1 = x 2 + 6x + 9 + x + 3 – 1 = x 2 + 7x + 11 Ex. 3 If f(x) = 4x and g(x) = 2x – 1, find g[f(5)]. g[f(5)] = g[4(5)] = g(20) = 2(20) – 1 = 39

81. 7.3 – Square Root Functions & Inequalities

82. Ex. 1 Identify the domain & range of each function. a. y = √ x + 4

83. Ex. 1 Identify the domain & range of each function. a. y = √ x + 4 x + 4 = 0

84. Ex. 1 Identify the domain & range of each function. a. y = √ x + 4 x + 4 = 0 x = -4

85. Ex. 1 Identify the domain & range of each function. a. y = √ x + 4 x + 4 = 0 x = -4 Domain: { x | x > -4}

86. Ex. 1 Identify the domain & range of each function. a. y = √ x + 4 x + 4 = 0 x = -4 Domain: { x | x > -4} y = √ x + 4

87. Ex. 1 Identify the domain & range of each function. a. y = √ x + 4 x + 4 = 0 x = -4 Domain: { x | x > -4} y = √ x + 4

88. Ex. 1 Identify the domain & range of each function. a. y = √ x + 4 x + 4 = 0 x = -4 Domain: { x | x > -4} y = √ x + 4 y = √ -4+ 4

89. Ex. 1 Identify the domain & range of each function. a. y = √ x + 4 x + 4 = 0 x = -4 Domain: { x | x > -4} y = √ x + 4 y = √ -4+ 4 y = 0

90. Ex. 1 Identify the domain & range of each function. a. y = √ x + 4 x + 4 = 0 x = -4 Domain: { x | x > -4} y = √ x + 4 y = √ -4+ 4 y = 0 Range: { y | y > 0}

91. Ex. 2 Graph each function. State the domain & range. a. y = √ x + 4 Domain: { x | x > -4}, Range: { y | y > 0} Graph: Y= 2 nd, x 2 x + 4) Zoom:6 2 nd Graph Plot at least 3 points of curve (x & y ints. & one other pt.)

92. xy -40 -31 02

93. Ex. 3 Graph each inequality a. y <√ x + 4 Graph: Y= Cursor left to \ Press “Enter” until (If > make it ) 2 nd, x 2 x + 4) Zoom:6 2 nd Graph Plot at least 3 points of curve (x & y ints. & one other pt.)

94. xy -40 -31 02