2. Function 1 st day

3. Objectives: At the end of the class, the learners shall be able to; 1. demonstrate an understanding of key concepts of functions. 2. accurately construct mathematical models to represent real-life situations using functions. 3. solves problems involving functions.

4. Think about this! Without using calculator, compute the value of expression by performing the operations indicated. InputRelationshipOutput 1. x = 2 2x+26 2

5. Think about this! Without using calculator, compute the value of expression by performing the operations indicated. InputRelationshipOutput 2. x = -2 2x+3 -2

6. Think about this! Without using calculator, compute the value of expression by performing the operations indicated. InputRelationshipOutput 3. x = 3 3x-36 3

7. Think about this! Without using calculator, compute the value of expression by performing the operations indicated. InputRelationshipOutput 4. x = 3 3(2x-3)9 3

8. https://www.youtube.com/watch?v=52tpYl2tTqk

9. Function is a relation where each element in the domain is related to only one value in the range by some rule. Three main parts: Input, relationship, and output What is function?

10. Input Function name What to output

11. Ordered Pairs (input, output) or (x, f(x)) = (x,y)

12. Domain, Codomain and Range The set "X" is called the Domain, The set "Y" is called the Codomain, and The set of elements that get pointed to in Y (the actual values produced by the function) is called the Range.

13. 2. Exactly one..." means that a function is single valued. It will not give back 2 or more results for the same input. How do we know the relationship is a function? 1. Each element..." means that every element in X is related to some element in Y. We say that the function covers X (relates every element of it). (But some elements of Y might not be related to at all, which is fine.)

14. Mapping of elements

15. “one-to-many” This is not ok in a FUNCTION ABAB CDCD ABAB CDCD "One-to-many" is not allowed, but "many- to-one" is allowed “many-to-one” This is ok in a FUNCTION X Y X Y

16. Remember: When a relationship does not follow those two rules then it is not a function... it is still a relationship, just not a function. All functions are relation but not all relations are functions.

17. Determine whether each mapping diagrams defines a function or a mere relation. 1. relation 2. function

18. Set of ordered pairs

20. Example: {(2,4), (2,5), (7,3)} is not a function because {2,4} and {2,5} means that 2 could be related to 4 or 5. In other words it is not a function because it is not single valued

21. Determine whether the given set of ordered pairs is a function or a mere relation. function { ( 1, 2), (2,3), (3,4), (4,5)} 1. relation { ( 1, 2), (1,3), (2,2), (2,3)} 2.

22. Graph of the relation

23. Vertical Line Test A graph represent a function if and only if each vertical line intersects the graph at most once. However, if a vertical line crosses the relation more than once, the relation is not a function.

26. not a function function

27. Tables of values

28. A function is a relation in which every input has exactly one output. Every x has exactly one corresponding y.

29. not a function function

30. not a function

31. Equation of the relation

32. When you are given an equation and a specific value for x, there should only be one corresponding y-value for that x-value. It is relatively easy to determine whether an equation is a function by solving for y.

33. function not a function

34. A. Determine if the following relations are functions. Then state the domain and range. 1. {(1, -2), (-2,0), (-1,2)} Function: _________ Domain: __________ Range: ___________ 2. {(1, 1), (2,2), (3,4)} Function: _________ Domain: __________ Range: ___________

35. B. Determine if the graph is function, then state the domain and range. Function: _________ Domain: __________ Range: ___________ https://www.dublin.k12.ca.us/cms/lib/CA0100142 4/Centricity/Domain/172/Function%20worksheet. pdf

36. Answer: A. 1. {(-1, -2), (-2,0), (-1,2)} Function: No Domain: {-1,-2,-1} Range: {-2,0,2} 2. {(1, 1), (2,2), (3,4)} Function: Yes Domain: {1,2,3} Range: {1,2,4}

37. Answer: B. Function: Yes Domain:{-8,2} Range: {-5,8}

38. What are the ways to tell if something is a function? Examining ordered pairs Solving for Y Vertical Line Test Mapping of elements Using an Input-Output Chart

40. Function Notation and Evaluation 2 nd day

41. #1 Checking of assignment September 11, 2019 Total of Item: 9

42. A. Determine if the following relations are functions. Then state the domain and range. 1. {(1, -2), (-2,0), (-1,2)} Function: _________ Domain: __________ Range: ___________ 2. {(1, 1), (2,2), (3,4)} Function: _________ Domain: __________ Range: ___________

43. B. Determine if the graph is function, then state the domain and range. Function: _________ Domain: __________ Range: ___________ https://www.dublin.k12.ca.us/cms/lib/CA0100142 4/Centricity/Domain/172/Function%20worksheet. pdf

44. Answer: A. 1. {(1, -2), (-2,0), (-1,2)} Function: Yes Domain: {-1,-2,-1} Range: {-2,0,2} 2. {(1, 1), (2,2), (3,4)} Function: Yes Domain: {1,2,3} Range: {1,2,4}

45. Answer: B. Function: Yes Domain:{-8,2} Range: {-5,8}

46. What are the ways to tell if something is a function? Examining ordered pairs Solving for Y Vertical Line Test Mapping of elements Using an Input-Output Chart

47. Review: Determine whether each mapping diagrams defines a function or not a function. Not a functionfunction

48. Review: Determine whether each table and graph defines a function or not a function. Not a function b

49. Review: Determine whether each statements defines a function or not a function. 1. y=3x 2. y=-2x+3 3. y= 20/x function Not a function

50. Awesome! 9= excellent 8-7 = Very good 6 = Good 5-4 = Satisfactory 3-2= Sufficient 1-0= Needs improvement

51. Function notation is the way a function is written. It is meant to be a precise way of giving information about the function without a rather lengthy written explanation. Function Notation

52. The most popular function notation is f (x) which is read "f of x". This is NOT the multiplication of f times x. f(x) and y mean the same thing. Traditionally, functions are referred to by single letter names, such as f, g, h and so on. Any letter(s), however, may be used to name a function.

53. Example: T he f (x) notation is another way of representing the y-value in a function, y = f (x). The y-axis may even be labeled as the f (x) axis, when graphing. Ordered pairs may be written as (x, f (x)), instead of (x, y).

54. Advantages of function notation: 1.It allows for individual function names to avoid confusion as to which function is being examined. Names have different letters, such as f (x) and g (x). The graphing calculator does distinctive function naming with Y1, Y2,... 2. It quickly identifies the independent variable in a problem. f (x) = x + 2b + c, where the variable is "x". 3.It quickly states which element of the function is to be examined. Find f (2) when f (x) = 3x, is the same as saying, "Find y when x = 2, for y = 3x."

56. Evaluating Function 2 rd day

57. To evaluate a function, substitute the input (the given number or expression) for the function's variable (place holder, x). Replace the x with the number or expression. Evaluating Function

58. Examples: Given the function f (x) = 3x - 5, find f (4). Solution: Substitute 4 into the function in place of x. f(x) = 3x-5 f (4) = 3(4) - 5 = 7. This answer can be thought of as the ordered pair (4,7). The answer may also be referred to as the image of 4 under f (x).

59. Evaluating Function vs Operation with function f(x+a) is not the same as f(x) + f(a) f(x+a) ≠ f(x) + f(a) (f+a)(x) is the same as f(x) + f(a) (f+a)(x) = f(x) + a(x)

60. Examples:

61. I recommend putting the substituted values inside parentheses (), so you don't make mistakes.

62. 1. The function f is defined on the real numbers by f(x) = 2 + x − x 2. What is the value of f(-3)? a. -10 b. 8 c. -4 d. 14 a

63. 2. The function g is defined on the real numbers by g(x) = (x 2 + 1)(3x − 5). What is the value of g(4)? a. -51 b.119 c. 63 d. 75 b

64. 3. Evaluate the function f(x) = |x − 5| for x = 3. a. -8 b.-2 c. 2 d. 8 c

65. 4. Evaluate the function g(x) = x 2 − 3x + 2 for x = a − 2. a. a 2 − 3a + 12 b. a 2 − a + 12 c. a 2 − 7a d. a 2 − 7a + 12 d

66. 5. f(x) = -2x 3 + ax 2 + 2, You are told that f(2) = 14, can you work out what "a" is? a. a = 6 b. a = 7 c. a = 8 d. a = 9 b

67. First evaluate f(2): f(2) = -2 × 2 3 + a × 2 2 + 2 = -16 + 4a + 2 = -14 + 4a But we are told f(2) = 14 So -14 + 4a = 14 Add 14 to both sides: So 4a = 14 + 14 = 4a = 28 ⇒ a = 28 ÷ 4 = 7

68. Operation with Function 3 rd day

69. We can add, subtract, multiply and divide functions! The result is a new function. Operation with Function

70. Let us try doing those operations on f(x) and g(x): Addition We can add two functions: (f+g)(x) = f(x) + g(x) Note: we put the f+g inside () to show they both work on x.

72. Subtraction We can also subtract two functions: (f-g)(x) = f(x) − g(x)

73. Multiplication We can multiply two functions: (f·g)(x) = f(x) · g(x)

75. Take a closer look!

81. Function Composition 4 th day https://www.google.com/url?sa=i&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwiaiaDq1 sbkAhUkKqYKHf7ADboQMwhUKAUwBQ&url=http%3A%2F%2Fwww.megcraig.org%2Ftag%2Ffunct ions%2F&psig=AOvVaw1i2JTHM8A9DY4FKBJC2-7g&ust=1568219448307561&ictx=3&uact=3

84. Example: f(x) = 2x+3 and g(x) = x 2 "x" is just a placeholder. To avoid confusion let's just call it "input": f(input) = 2(input)+3 g(input) = (input) 2

85. Let's start: (g º f)(x) = g(f(x)) First we apply f, then apply g to that result: (g º f)(x) = (2x+3) 2 What if we reverse the order of f and g? * (f º g)(x) = f(g(x))

86. The result of f(x) = 2x+3 and g(x) = x 2 when we try to evaluate using composite function is now like this, (f ● g)(x) = f[g(x)] = 2x+3 = 2( x 2 )+3 (f ● g)(x)=2x 2 +3

88. Compute the following: 1. If f(x) = x 2 – 4x + 2 and g(x) = 3x – 7, find (f ● g)(x) 2. If g(x) = –6x + 5 and h(x) = –9x – 11, find (g ● h)(x) 3. If and g(x) = 5x 2 – 3, find (g ● f)(x) http://www.mesacc.edu/~scotz47781/mat120/not es/composition/composition_practice.html

89. Answer: 1.

90. 2.

91. 3.

92. Application of Function https://www.analyzemath.com/function/applicati ons.html

93. Examples of applications of functions where quantities such area, perimeter, chord are expressed as function of a variable.

94. Steps to Solving Word Problems Read through the problem and set up a word equation — that is, an equation that contains words as well as numbers. Plug in numbers in place of words wherever possible to set up a regular math equation. Use math to solve the equation. Answer the question the problem asks.

95. Problem 1: A right triangle has one side x and a hypotenuse of 10 meters. Find the area of the triangle as a function of x.

98. Problem 2: A rectangle has an area equal to 100 cm2 and a width x. Find the perimeter as a function of x.