Functions and it's graph6519105021465481791.pptx

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1. Functions and its Graph

2. Relation Objectives: Define a relations; identify domain and range of a relation.

3. WORD HUNT Find as many words as you can that can be associated on the topic Function and Relation.

4. WORD HUNT Find as many words as you can that can be associated on the topic Function and Relation. Function Relation Domain Range Set Table Mapping Graph Equation Input Output

5. Getting Ready To be able to save money and energy, just dial “RENT CAR”.

6. Relation is referred to as any set of ordered pair. Conventionally, It is represented by the ordered pair ( x , y ). x is called the first element or x-coordinate while y is the second element or y-coordinate of the ordered pair. DEFINITION Relation is a rule that relates values from a set of values (called the domain ) to the second set of values (called the range )

7. Ways of Expressing a Relation 5. Mapping 2. Tabular form 3. Equation 4. Graph 1. Set notation

8. . Example: Express the relation y = 2 x when x = 0,1,2,3 in 5 ways. 1. Set notation (a) S = { ( 0, 0) , ( 1, 2 ) , ( 2, 4 ), ( 3, 6) } or (b) S = { (x , y) / y = 2x, x = 0, 1, 2, 3 } 2. Tabular form x 1 2 3 y 2 4 6

9. 3. Equation: y = 2x 4. Graph y x 5 -4 -2 1 3 5 5 -4 -2 1 3 5 -5 -1 4 -5 -1 4 -3 -5 2 2 -5 -3 ● ● ● 1 2 2 4 6 3 x y 5. Mapping

10. DEFINITION: Domain and Range All the possible values of x is called the domain and all the possible values of y is called the range . In a set of ordered pairs, the set of first elements and second elements of ordered pairs is the domain and range, respectively. Example: Identify the domain and range of the following relations. 1.) S = { ( 4, 7 ),( 5, 8 ),( 6, 9 ),( 7, 10 ),( 8, 11 ) } Answer : D: { 4,5,6,7,8} R:{7,8,9,10,11}

11. 2.) S = { ( x , y ) / y = | x | ; x  R } Answer: D: all real nos. R: all real nos. > 3) y = x 2 – 5 Answer. D: all real nos. R: all real nos. > -5 4) | y | = x Answer: D: all real nos. > R: all real nos.

12. 5. 7. 8. 6. X 1 2 3 4 y 6 8 10 12 14 X 2 1 1 2 y -8 -4 4 8 X -4 -2 2 4 y 1/8 1/4 1/2 1 2 X 2 4 2 Y 1 1/2 1/2 1 Domain: {0, 1, 2, 3, 4} Range: {6, 8, 10, 12, 14} Domain: {0, 1, 2} Range: {-8, -4, 0, 4, 8} Domain: {-4, -2, 0, 2, 4} Range: {1/8, ¼, ½, 1, 2} Domain: {0, 2, 4} Range: {0, ½, 1}

13. A B C D E F 1 2 3 4 5 6 9. Star Square Circle Triangle Diamond Oblong 1 2 3 4 5 10. 2 3 A B C D E F 11. 12. 13. 14. Domain: {A,B,C,D,F} Range: {1, 2, 4, 5, 6} D: { star,square,circle,triangle , diamond, oblong} Range: {1, 2, 3, 4} Domain: {2, 3} Range: {A, B, C, D, E, F} Domain: { x|x  ℜ } Range: { y|y  ℜ } Domain: {x| x  ℜ , -2 ≤ x ≤ +2} Range: {y| y  ℜ , -2 ≤ y ≤ +2} Domain: { x|x  ℜ + } Range: { y|y  ℜ }

14. RememberThis ! Identifying Domain and Range 1. Set of Ordered Pairs: a. Domain – a set of all first coordinates of the ordered pairs. b. Range – a set of all second coordinates of the ordered pairs 2. Tabular Form: a. Domain – a set of all x-values. b. Range – a set of all y-values. 3. Mapping Diagram: a. Domain – a set of all elements on the first group. b. Range – a set of all elements on the second group. 4. Graphing: a. Domain – a set of all values on the x-axis covered by the graph. b. Range – a set of all values on the y-axis covered by the graph.

15. Let Us Try This! Determine domain and range of the following: S = {(a,2), (b,4), (c,6), (d,8), (e,10)} S = {(-6,9), (6,-9), (9,6), (-9,6)} 4. 6.

16. Let Us Try This! Determine the domain and range of the following . y = 4x 2 + 12 y = 14 - x 10.     2 3 4 5 6 7       1 3 5 2 4

17. Function Objectives: define intuitively a function; and identify real life examples of a function including piecewise functions;

18. Review: What is a relation? In a relation, how can you identify a independent variable and dependent variable? What is another name you can give to independent variable? Dependent ariable ? Give five examples of relation citing the independent variable and dependent variable. For each example, can you identify if the relation works or valid?

19. Definition: Function A function is a special relation such that every first element is paired to a unique second element. It is a set of ordered pairs with no two pairs having the same first element.

20. Real Life Examples of Function

21. Real Life Examples of Function

22. Real Life Examples of Function

23. Real Life Examples of Function

24. Real Life Examples of Function

25. Real Life Examples of Function

26. Piecewise or Split Function

27. Piecewise or Split Function

28. Real Life Examples of Piecewise or Split Function

29. Real Life Examples of Piecewise or Split Function

30. Real Life Examples of Piecewise or Split Function

31. Real Life Examples of Piecewise or Split Function

32. Let’s Get Real! Tell whether the following situation describes a function or not. The amount of postage placed on a first-class letter depends on the weight of the letter. The temperature of a cup of coffee depends on how big the cup is. The height which a football attains depends upon the number of seconds after it was kicked. The number of pages read is related to the number of words written on every page. The speed of the bullet depends on its acceleration. The amount in peso of a $100 bill is related to the country’s inflation rate. The overtime pay of an employee is related to the number of hours he worked overtime. The amount paid for a long distance call depends on how long the call is made. The amount of monthly internet bill depends on the amount of data being used for the post paid consumers. The amount of tax a company pays depends upon its net profit for the year.

33. How to Differentiate a Function from a Relation?

34. Differentiating Function from a Relation Objectives: differentiate a function from a relation; and

35. One-to-one and many-to-one functions Each value of x maps to only one value of y . . . Consider the following graphs Each value of x maps to only one value of y . . . BUT many other x values map to that y. and each y is mapped from only one x. and Functions

36. One-to-one and many-to-one functions is an example of a one-to-one function is an example of a many-to-one function Consider the following graphs and Functions One-to-many is NOT a function. It is just a relation. Thus a function is a relation but a relation could never be a function.

37. Identify whether the given set is a function or just a merely a relation. A = {(-1,-1), (0,1), (1,0), (1,1)} B = {(-1,-1), (0,1), (1,0), (2,-2)} C = {(-1,-1), (0,1), (1,0), (2,-1)} D = {(-1,-1), (0,1), (1,0), (0,-1), (1,-1)} Not a Function Function Function Not a Function

38. 2. Identify whether the given table is a function or just a merely a relation. Function Not a Function Function Not a Function X 1 2 3 4 y 6 8 10 12 14 X 2 1 1 2 y -8 -4 4 8 X -4 -2 2 4 y 1/8 1/4 1/2 1 2 X 2 4 2 Y 1 1/2 1/2 1

39. 3. Identify whether the following graph is a function or just a merely a relation. Function Not a Function Function Not a Function Function Function Function

40. Vertical Line Test – an imaginary line used to test whether the graph is a function or not. The VLT intersect the graph at only 1 point, therefore it is a FUNCTION The VLT intersect the graph at more than 1 point, therefore it is NOT a FUNCTION

41. Many-to-one 4. Identify whether the given illustration describes a function or just a merely a relation . Noel Joel Kim Julie Nina Joy Bird Cat Dog Turtle Mice Pig Star Square Circle Triangle Diamond Oblong 1 2 3 4 5 A B C D E F 1 2 3 4 5 6 2 3 A B C D E F Function Function Not a Function Not a Function One-to-one Many-to-many One-to-many

42. 5. Identify whether the given equation is a function or just a merely a relation. a. y = 4x + 8 b. 4x 2 + 9y 2 = 16 c. y = 2x 2 – 3x + 16 d. x = 3y 3 + 10 e. y = f. y = Not a Function Function Function Not a Function (if the range is a set of real numbers) Not a Function Function Function (if the range is a set of positive real numbers)

43. How to Determine a Function Using the Five Ways in Representing a Relation? By Set Notation - There should be no duplication on the first coordinate By Tabular Form – There should be no duplication on the x -values. By Graphing – The Vertical Line Test (VLT) should intersect the graph at only one point. By Mapping ( Arrow Diagram ) – One-to-one Correspondence/ Many-to-one Correspondence By Equation – The exponent of the dependent variable ( y ) is one or odd.

44. Let Us Try This! Determine whether the following is a function or just a mere relation. S = {(a,2), (b,4), (c,6), (d,8), (e,10)} S = {(-6,9), (6,-9), (9,6), (-9,6)} 4. 6.

45. Let Us Try This! Determine whether the following is a function or just a mere relation. y = 4x 2 – 2x + 12 x 2 – 3y 2 = 14 10.     2 3 4 5 6 7       1 3 5 2 4

46. Identify which of the following relations are functions. 1) S = { ( 4, 7 ), ( 5, 8 ), ( 6, 9 ), ( 7, 10 ), ( 8, 11 ) } 2) S = { ( x , y ) / y = | x | ; x   } 3) y = x 2 – 5 4) | y | = x 5) x 2 – 4x = y 2 – 3x 8. 6) 7) X 1 -1 11 -1 y 12 4 -6 4 X 1 -1 11 -11 y 4 44 -44 9.

47. Identify which of the following relations are functions. A B C D E 2 3 4 5 W X Y Z 1 3 5 2 4 10. 11. 12. 13.

48. Evaluating Functions Objectives: Evaluate a function given a value of x.

49. Meet best friends Emily and Anton. They have formed a partnership. Anton, being the creative one, makes costume jewelry. Emily, being business-minded, markets the jewelry Anton makes. In a recent month, they spent Php1,000 on raw materials to make 50 pieces of jewelry and sell each for Php25.00. assuming that they are not required to pay a sales tax, their net profit depends on the number of pieces of jewelry sold. The problem includes three constants: the fixed cost of raw materials (Php1,000), the selling price for each piece (Php25.00), and the total number of jewelry pieces made (50). SITUATION:

50. SITUATION: Number of Pieces of Jewellery Sold (n) Net Profit in Pesos ( Php ) -1,000 10 -750 20 -500 25 -375 30 -250 45 125 Let us use n to represent the number of pieces of costume jewelry sold and P for the net profit in pesos. a. What should be the greatest value of n? The greatest value of P? b. What should be the least value of n? The least value of P? Some of the data for selling the costume jewelry is shown in the table at the right.

51. SITUATION: Show the function machine that clearly shows the input, the process, and the output after selling the costume jewelry is given as follows. Write the equation that shows algebraically how to compute the net profit given the number of pieces of jewelry sold. If there are 40 pieces of jewelry sold, how much should be the profit? Number of jewelry (n) x 25 - 1000 f(x) = 25x - 1000

52. THE FUNCTION NOTATION: The name of a function is f . Other letters may also be used to name a function. f(x) is read as “f of x”, and this represent the value of the function at x. The function notation y = f(x) tells you that y is a function of x . if there is a rule relating y to x , such as y = 3 x +1, then you can also write: f(x) = 3x + 1

53. DEFINITION: Function Notation Letters like f , g , h and the likes are used to designate functions. When we use f as a function, then for each x in the domain of f , f ( x ) denotes the image of x under f . The notation f ( x ) is read as “ f of x ”.

54. EXAMPLE: Evaluate each of the following function. 1. If f(x) = x + 8, find: a. f(4) b. f(-2) c. f(-x) d. f(x + 3) e. f(x 2 - 3) 2. If g(x) = x 2 – 4x, find: a. g (-5) b. g (0) c. g (4) d. g (2x) e. g (x - 1)

55. MORE EXERCISES: Evaluate each of the following function. 1. If f ( x ) = x + 9 , what is the value of f ( 6 ) ? 2. If g ( x ) = 2x – 12 , what is the value of g (– 2 )? 3. If h ( x ) = x 2 + 5 , find h ( x + 1 ). 4. If h ( t ) = 3t + 5 and t(x) = x 2 + 6, find h(x). 5. If k ( t ) = 6t + 5, t(s) = ( s + 1 ) and s = 9, find k( t ) and t( s ).

56. ASSESSMENT: Evaluate each of the following function. 1. If f(x) = 9 – 6x , find: a. f(-1) b. f(2/3) c. f(-3+x) d. f(2x-1) 2. If Given f(x) = , find a. f(2) b. f(0) c. f(2 – x) d. f(a + b)  

57. Evaluating Piecewise Functions Objectives: Evaluate a piecewise function given a value of x.

58. Piecewise Defined Function if x < 0 if A piecewise function is defined by different formulas on different parts of its domain. Example: ; x < 2 ; x = 2 ; x > 2

59. You plan to sell cupcakes to raise funds. A bakery charges you Php 15.00 for the first 100 cupcakes. After the first 100 cupcakes you purchase up to 150 cupcakes, the bakery will lower the price to Php13.00 per cupcake. After you purchase 150 cupcakes, the price will decrease to Php 10.00 per cupcake. write the piecewise function that describes what the bakery charges on the cupcakes. Graph the function SITUATION:

60. Piecewise Defined Function if x<0 Find the value of the following: f(-2), f(-1), f(0), f(1), f(2) EXAMPLE: Evaluate the piecewise function at the indicated values. if find the value of the following: f(-5), f(0), f(1), f(5) if if if

61. Let’s Try This! Find the value of the following: f(-5), f(-1), f(0), f(1), f(5) find the value of the following: f(-5), f(0), f(1), f(5) if x > 2 if -2 ≤ x ≤ 2 if x < -2

62. Evaluate each function at the indicated values of the independent variable and simplify the result. 1. Given f(x) = 9 – 6x, find a. f(-1) b. f(2/3) c. f(-3 + x) d. f(2x – 1) 2. Given f(x) = , find a. f(2) b. f(0) c. f(2 – x) d. f(2a) 3.

63. Operations on Functions Addition of Functions If f and g are functions, their sum is a function defined as (f + g)(x) = f(x) + g(x) Subtraction of Functions If f and g are functions, their difference is a function defined as (f - g)(x) = f(x) - g(x) Multiplication of Functions If f and g are functions, their product is a function defined as (f ● g)(x) = f(x) ● g(x) Division of Functions If f and g are functions, their quotient is a function defined as (f ÷ g)(x) = f(x) ÷ g(x)

64. Let’s try this! Given the following functions f(x) = 2x g(x) = 4x – 5 h(x) = x 2 + 1. Find the following: a. (g + h)(x) b. (h - g)(x) c. (h ● h)(x) d. (g ● h)(x) e. (g ÷ f)(x) f. (h ÷ g)(x)

65. Let’s try this! 1. If f(x) = 2x 2 – 1 and g(x) = x 2 + x, find: a. (f + g)(x) b. (f – g)(x) c. (f ● g)(x) d. (f ÷ g)(x) 2. If k(x) = 3x 2 – 5 and h(x) = 2x + 6, find: a. (k + h)(x) b. (k – h)(x) c. (k ● h)(x ) d. (k ÷ h)(x)

66. Given f(x) = 11– x and g(x) = x 2 +2x –10 evaluate each of the following functions. f(-5) g(2) (f g)(5) (f - g)(4) f(7)+g(x) g(-1) – f(-4) (f ○ g)(x) (g ○ f)(x) (g ○ f)(2) (f ○ g) Let Us Try This!

68. Composition of Functions Composite Functions If f and g are functions, the composite function f with g is defined as (f ○ g)(x) = f(g(x)) The domain of (f ○ g)(x) is the set of all x such that a. x is the domain of g ; and b. g(x) is the domain of f . Let f(x) = √x, g(x) = 4x 2 – 5x, and h(x) is x + 1. Find the following: a. (g ○ h)(x) b. (h ○g)(x) c. [h ○ h](x) d. [f ○ (g + h)](x)

69. Let’s Try This!

70. Given f(x) = x 2 – 2 and g(x) = x + 2. Find: a) (f ○ g)(x) b) (g ○f)(x) c) (f ○ g)(-1) d) (g ○f)(2) Given f(x) = x - 8 and g(x) = Find: a) (f ○ g)(x) b) (g ○f)(x) c) (f ○ g)(0) d) (g ○f)(1)

71. Given f(x) = 3x – 1, g(x) = x 2 + 1, and h(x) = . Find: a. (f + g)(2) b. (g – f)(x) c. (f ●g)(x) d. (g ÷ f)(1) e. (h ○ f)(-1) f. (h ○ g)(0) g. (f ○ g)(x) h. (g ○ f)(c)

73. Problem Solving Steps in Problem Solving: 1. READ the problem carefully and draw the picture that conveys the given information. Identify what information is given. Identify what you are asked to find. Choose a variable to represent one of the unspecified number in the problem. 2. PLAN the solution . After defining the variables, find a word sentence to suggest an equation for the number(s). use the expressions to replace the word sentence by an equation.

74. Problem Solving Steps in Problem Solving: SOLVE the problem . To solve the equation, familiarize yourself with the properties of equality and to use PEMDAS rule for series of operations. 4. EXAMINE the solution . Use the solution of the equation to write a statement that settles the problem. Check that the conclusion agrees with the problem situation or satisfies all conditions of the problem.

75. Problem Solving Example: 1. The perimeter of a rectangle is 100 cm. express the area of the rectangle in terms of the width x . 2. A piece of wire x cm long is bent into the shape of a circle. Express the area ( A ) of the circle in terms of x , the circumference. 3. Express the length of the radius of a circle as a function of the area of the circle.

76. Problem Solving Example: 4 . Mang Juan wanted to build a fence for his ducks beside the river. The fence is rectangular in shape with an open side on the river. What is the maximum area made by 100 meters of fencing? 5. The length of a rectangle is twice its width. When the length is increased by 5 and the width is decreased by 3, the new rectangle will have a perimeter of 52. Find the dimensions of the original rectangle?

77. Problem Solving Let’s Try This: 1. The perimeter of a rectangle is 24 cm. Express the area of the rectangle in terms of the width x. 2. The area of a rectangle is 85 cm 2 . Express the perimeter of the rectangle as a function of the width x. 3. A piece of wire y cm long is bent into a circle. Express the area of the circle as a function of the circumference y.

78. Problem Solving Let’s Try This: 4 . Two years ago, Nelly was three times as old as her nephew was then. In five years, Nelly will be only two times as old as her nephew. How old is each? 5. The height of a projectile fired upward is given by the formula S = vt – t 2 , where S is the height, v is the initial velocity, and t is the time. Find the time for a projectile to return to earth if it has an initial velocity of 98 mps.

79. E V A L U A T I O N Problem Solving: 1a. The perimeter of a rectangle is 160 cm. Express the area of a rectangle in terms of width, x. b. What is the area of a rectangle if its width is 8 cm? 2a. A piece of wire, x cm long is bent into the shape of a circle. Express the area of the circle in terms of x. b. If the wire is 20 cm long, what would be the area of a circle made by the wire?

80. E V A L U A T I O N 1a. The perimeter of a rectangle is 100 cm. Express the area of a rectangle in terms of width, x. b. What is the area of a rectangle if its width is 6 cm? 2a. A piece of wire, x cm long is bent into the shape of a circle. Express the area of the circle in terms of x. b. If the wire is 314 cm long, what would be the area of a circle made by the wire? 3a. The area of a rectangle is 1601 cm 2 . Express the perimeter of the rectangle as a function of the width, x. b. If the width of a rectangle is 10 cm, Find the perimeter.

81. E V A L U A T I O N 1a. The area of a rectangle is 400 cm 2 . Express the perimeter of a rectangle in terms of length, x. b. What is the perimeter of a rectangle if its length is 24 cm? 2a. Express the circumference of a circle in terms of its area. b. If the area of a circle is 31,416cm 2 , what would be its circumference? 3. Harry is 20 years older than Ivan. Five years from now, Ivan will be 3/5 as old as Harry. How old is each now?