1. B. Functions Calculus 30

2. 1. Introduction A relation is simply a set of ordered pairs. A function is a set of ordered pairs in which each x-value is paired with one and only one y-value. Graphically, we say that the vertical line test works.

5. Note*

6. We use “function notation” to substitute an x-value into an equation and find its y-value

7. Examples

8. Assignment Ex. 2.1 (p. 55) #1-10

9. 2. Identifying Functions

10. Example

11. The polynomials function has degree “n” (the largest power) and leading coefficient

12. Example

13. A polynomial function of degree 0 are called constant functions and can be written f(x)=b Slope = zero

14. Example

15. Polynomial functions of degree 1 are called linear functions and can be written y = mx + b m= slope b= y-intercept Example Graph

18. Polynomial functions of degree 3 are called cubic functions Example Graph Degree 4 functions with a negative leading coefficient Example Graph Degree 5 functions with a negative leading coefficient Example graph

19. Summary: Polynomial functions of an odd degree and positive leading coefficient begin in quadrant 3 and end in quadrant 1 Polynomial functions of an odd degree and negative leading coefficient begin in quadrant 2 and end in quadrant 4 Polynomial functions of an even degree and positive leading coefficient begin in quadrant 2 and end in quadrant 1 Polynomial functions of an even degree and negative leading coefficient begin in quadrant 3 and end in quadrant 4

22. Examples

23. Notice that all the graphs pass through the points (0,0) and (1,1). This is true for all power functions

28. Any x-value which makes the denominator = 0 is a vertical asymptote. If degree of p(x) < degree of q(x), there is a horizontal asymptote at y=0 (x-axis) If degree of p(x) = degree of q(x), there is a horizontal asymptote at y = k, where k is the ratio of the leading coefficients of p(x) and q(x) respectively.

29. Example

31. Examples

32. Thus the graphs of algebraic functions vary widely.

35. Graphs of exponential functions always pass through (0,1) and lie entirely in quadrants 1 and 2 If b>1, the graph is always increasing and if 0<b<1, the graph is always decreasing. The x axis is a horizontal asymptote line.

36. Example

38. Graphs of log functions always pass through (1,0) and lie in quadrants 1 and 4 If b>1, the graph is always increasing and if 0<b<1, the graph is always decreasing. The y-axis is a vertical asymptote line.

39. Examples

40. h) Transcendental Functions are functions that are not algebraic. They included the trig functions, exponential functions, and the log functions

41. Assignment Ex. 2.2 (p. 64) #1-4

42. 3. Piecewise and Step Function a) A Piecewise Function is one that uses different function rules for different parts of the domain. Watch open and closed intervals and use corresponding dots To find values for the function, use the equation that contains that value (on the graph) in its domain.

43. Example

46. Examples

48. Example Graph

49. Using your graphing Calculator

51. Examples

52. In other words, this function rounds non-integer values up and is called the least integer function or ceiling function

53. Assignment Ex. 2.3 (p. 70) #1-5

54. 4. Characteristics of Functions a) A function is said to be even if it is symmetrical around the y- axis. That is, f(x) and f(-x) are the same value

55. Examples

56. Notice that every point (a,b) is the 1 st quadrant has a mirror image, (-a,b) in the second quadrant

58. Examples

59. Notice that every point (a,b) has a corresponding point (-a, -b)

60. Can a function be both even and odd? Explain/Prove.

62. Example

63. d) A function is one-to-one if neither the x nor the y-values are repeated Examples A function is many-to-one if y-values are repeated Examples

64. What is mapping notation?

65. Can a function be one-to-many? Why or why not?

66. Assignment Ex. 2.4 (p. 79) #1-9

67. 5. Graphing Transformations a) Vertical Shirts – simply add “c” to shift up “c” units and subtract “c” to shift graph down “c” units

68. Examples

69. b) Horizontal Shifts – for f(x), f(x+c) will shift the graph “c” units to the left and f(x-c) will shift the graph “c” units to the right

70. Examples

71. c) Vertical Stretches – for f(x), c(f(x)) where c>1, will stretch the graph vertically by “c” units That is, all the y-values are “c” times higher than before (multiply the y by c)

72. Examples

75. e) Horizontal Compressions – for f(x), f(cx), where c>1, will compress the graph horizontally by c units. That is, the function reaches its former y-values c times sooner. (divide x by c)

76. Examples

82. Example

86. Assignment Ex. 2.5 (p. 90) Oral Ex. 1-15 Written 1-36 odds

87. 6. Finding Domain and Range a) The Domain (x-values) and Range (y-value) may be determined b examining the graph of the function

88. Examples

89. b) The domain and range of the function can also be determined by examining the equation of a function.

90. You analyze the equation for restrictions on the domain. That is, are there any x-values that would make a denominator equal to zero or a negative value under an even root sign. Generally, restrictions on the domain will cause restrictions of the range.

91. Example

92. Examples

93. Recall that you cannot find the logarithm for a non-positive number

94. Example

95. Domain Summary You cannot divide by zero. You cannot take the even root of a negative number. You cannot find the logarithm of a non-positive number.

96. Finding The Range There is no rule for finding the range of a function. Generally students need to be asking themselves questions such as: What happens to the value of the function for large positive x values? What happens to the value of the function for large negative x values? What happens to the value of the function near to any values in the domain that cause the denominator of the function to be zero? Do the numerator, denominator, or any part of the expression ever reach a minimum/maximum value? Determining the horizontal and vertical asymptote lines (Math B30) together with a sign analysis is helpful for rational functions.

97. Assignment Ex. 2.6 (p. 99) #1-45 odds

98. 7. Combinations of Functions

99. Example

105. Examples

107. Assignment Ex. 2.7 (p. 106) #2-16