1. Ch1.1 – Functions Functions – for every element x in a set A function machine there is exactly one element y from set B x that corresponds to it. Set A – (the set of inputs) is the domain y Set B – (set of outputs) is the range Ex1) A = {a,b,c}, B = {1,2,3,4,5}. Which of following are functions from A to B? a) {(a,2),(b,2),(c,4)}b) {(a,4),(b,5)} c)d) f(x) abcabc 1234512345 abcabc 1234512345

2. y = x 2 dependent independent variable f(x) = x 2 Ex2) Which represent y as a function of x? a) x 2 + y = 1b) –x + y 2 = 1

3. Ex3) Let g(x) = -x 2 + 4x + 1 Solve: a) g(2)b) g(t)c) g(x+2) Ex4) Evaluate the piecewise function for x = -1,0,1 x 2 + 1 x < 0 x – 1 x > 0 f(x) =

4. Ex5) Find the domain of each: a) f:{(-3,0),(-1,4),(0,2),(2,2),(4,1)} ChP.1A p92 1-7odd,25-41odd (just a and c)

5. Ch1.1A p92+ 1-7odd,25-41odd

11. ChP.1B – More Functions Ex1) Find all real values for x such that f(x) = 0 in: Ex2) Find the values where f(x) = g(x): f(x) = x 2 + 2x + 1g(x) = x + 2

12. Ex3) For f(x) = x 2 – 4x + 7, find

13. HW#75) For g(x) = 3x – 1 find: HW#76) Ch1.1B p92+26-40even, 43-59odd,71-75odd

14. Ch1.1B p92+ 26-40even,43-59odd,71-75odd

20. Ch1.2 – Graphing Functions 4 Ex1) The graph of function f is shown. 3 a) Find the domain. 2 b) Find the values of f(-1) and f(2) 1 c) Find the range of f. -4 -3 -2 -1 -1 1 2 3 4 5 -2 -3 -4 -5 (-1,-5) (2,4) (4,0)

21. Ex2) Use the vertical line test to determine which graphs represent y as a function of x. a)b)c)

22. Increasing Function – if x 1 < x 2, then f(x 1 ) < f(x 2 ) Decreasing function – if x 1 f(x 2 ) Constant function – for all x, f(x 1 ) = f(x 2 ) Ex3) For the following graphs determine where the functions are increasing, decreasing, and constant. t + 1 t < 0 a) f(x) = x 3 b) f(x) = x 3 – 3x c) f(t) = 1 0 < t < 2 -t + 3 t > 2

23. Symmetry A function f is even if for each x: f(x) = f(-x) A function f is odd if for each x: f(-x) = -f(x) symmetry to y-axis symmetry to origin symmetry to x-axis

24. Ex4) Determine whether each function is odd, even, or neither : a) g(x) = x 3 – xb) h(x) = x 2 + 1c) f(x) = x 3 – 1 Ch1.2 p105+ 1-13odd,19,21, 25-33odd,37-45odd

25. Ch1.2 p105 1-13odd,19,21,25-33odd,37-45odd

32. Ch1.3 – Graphs of Functions The most common graphs in algebra: f(x) = c f(x) = x f(x) = |x| Constant functionIdentity function Absolute value function f(x) = f(x) = x 2 f(x) = x 3 Square root functionSquare function cube function (works w all even powers) odd powers)

33. Shifts: Shift upward: f(x) + c Shift downward: f(x) – c Shift right: f(x – c) Shift left: f(x + c) Ex1) How does each function compare to f(x) =x 3 a) g(x) = x 3 + 1b) h(x) = (x – 1) 3 c) k(x) = (x + 2) 3 + 1

34. Ex2) Find eqns for each function: f(x) = ___ g(x) = ___ h(x) = ___ g(x) h(x) f(x) 6 5 4 3 2 1 -2 -3 -4 -5 -6 -6 -5 -4 -3 -2-1 1 2 3 4 5 6

35. Reflections: f(x) = x 2 Reflection to the x-axis: Reflection to the y-axis

36. Ex3) Find eqns for each function that is a transformation of f(x) =x 4 a) g(x) = b) h(x) = Ch1.3A p116+ 1-9odd 13,19,23

37. Ch1.3A p116 1-9odd,13,19,23

39. 23.

40. Ch1.3B – More Graphs Ex4) Graph each: a) f(x) = – b) g(x) = c) h(x) = –

41. Nonrigid Transformations – stretch and shrink graphs Ex5) Compare each function to f(x) = |x| g(x) = 3|x|h(x) =

42. Ex6) Use a calculator to graph: g(x) = 5(x 2 – 2)h(x) = 5x 2 – 2 Ch1.3B p116+ 15,17,21,25-35odd,41,43

43. ID the common function and what transformation is shown

44. Ch1.3B p116+ 15,17,21,25-35odd,41,43

46. Ch1.1 – 1.3 Mid Chapter Review Ch1.1 p92+ 52,54,58,72,74 Ch1.2 p105 2,4,6,10,20,22,28,30 Ch1.3 p116 2,4,10,16,18,20, 26-36even

47. Ch1.1 p92+ 52,54,58,72,74

48. Ch1.2 p105 2,4,6,10,20,22,28,30

49. Ch1.2 p105 2,4,6,10,20,22,28,30

50. Ch1.2 p105 2,4,6,10,20,22,28,30

52. Ch1.3 p116 2,4,10,16,18,20, 26-36even

53. Ch1.3 p116 2,4,10,16,18,20,26-36even

55. Ch1.4 – Combinations of Functions 1. Sum: (f + g)(x) = f(x) + g(x) 2. Difference: (f – g)(x) = f(x) – g(x) 3. Product: (f. g)(x) = f(x). g(x) 4. Quotient:

56. Ex1) Find (f + g)(x) for the functions f(x) = 2x+1 and g(x) = x 2 + 2x – 1 for x = 2. Ex2) Find (f – g)(x) for the functions f(x) = 2x+1 and g(x) = x 2 + 2x – 1 for x = 2

57. Ex3) Find and for the functions and list their domains.

58. HW#s: 11, 21, 23 in class

59. Ch1.4A p126 5 -19odd HW#s: 11, 21, 23 in class

60. Ch1.4A p 139 5 – 19 odd, (11,21,23 in class)

62. Ch1.4B – Composition of Functions Ex4) Find for f(x) = x > 0 g(x) = x – 1 x > 1 Solve for and if possible.

63. Ex5) Given f(x) = x + 2 and g(x) = 4 – x 2 find and

64. Ex6) Given f(x) = x 2 – 9 and g(x) = find and

65. Ex7) Given f(x) = 2x + 3 and g(x) = ½(x-3) find and

66. Ex8) Express h(x) = as a composition of two functions.

67. Ex9) The number of bacteria in your food once pulled from the fridge is given by N(T) = 20T 2 – 80T +500 2 < T < 14, where T is temp in ˚C. When food is removed from fridge, it temp changes with time (t) by: T(t) = 4t + 2 0 < t < 3, t in hours. a) What does the composite N(T(t)) represent? b) What is the # of bacteria at t = 2hrs? c) At what time does the bacteria count reach 2000? Ch1.4B p128 35-43odd, 49-53odd,61,63

71. Ch1.5 – Inverse Functions DomainRange RangeDomain 12341234 56785678

72. Ch1.5 – Inverse Functions DomainRange f(x) = x + 4 RangeDomain f -1 (x) = x – 4 Inverse functions have the effect of undoing each other. By defn, the domain of f must equal the range of f -1. Check with and 12341234 56785678

73. Ex1) Find the inverse of f(x) = 4x and verify that and equal the identity function.

74. Ex2) Find the inverse of f(x) = x – 6 and verify that and equal the identity function.

75. Ex3) Show that these functions are inverses: f(x) = 2x 3 – 1

76. Graphs of inverses are reflections around the y = x line. Ex4) Graph Ex3 functions: f(x) = 2x 3 – 1 Ch1.5A p139+ 5-10 all, 11-17odd

79. Ch1.5B – One-to-one Functions Not all functions have an inverse. To have an inverse a function must be one-to-one To test, graph func on calc. If passes vertical and horizontal line tests, then it has an inverse. Ex5) Have inverses? f(x) = 2x 3 – 1 g(x) = x 2 – x

80. Ex6) Find the inverse of Procedure: 1. Check w calc that it has inverse. 2. Replace f(x) with y. 3. Interchange x and y 4. Solve for y. 5. If it works, call it f -1 (x).

81. Ex7) Find the inverse of and sketch. Ch1.5B p139 21,22,26,28,41-55odd

85. Ch1 Rev p152 1 – 63 odd

93. 6 5 4 3 2 1 -2 -3 -4 -5 -6 -6 -5 -4 -3 -2-1 1 2 3 4 5 6