# x 2 if 2 x < 0 4 x if 2 x 6

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1 Piecewise-defined Functions Example Consider the function f defined by x if x < 0 f (x) = x if 0 x < 4 x if x 6

2 Piecewise-defined Functions Example Consider the function f defined by x if x < 0 f (x) = x if 0 x < Find the value of () f ( ) () f ( ) (3) f (.5) and sketch the graph of f. 4 x if x 6

3 Piecewise-defined Functions Example Consider the function f defined by x if x < 0 f (x) = x if 0 x < Find the value of () f ( ) = ( ) = () f ( ) (3) f (.5) and sketch the graph of f. 4 x if x 6

4 Piecewise-defined Functions Example Consider the function f defined by x if x < 0 f (x) = x if 0 x < Find the value of () f ( ) = ( ) = () f ( (3) f (.5) ) = = and sketch the graph of f. 4 x if x 6

5 Piecewise-defined Functions Example Consider the function f defined by x if x < 0 f (x) = x if 0 x < Find the value of () f ( ) = ( ) = () f ( ) = = (3) f (.5) = 4.5 =.5 and sketch the graph of f. 4 x if x 6

6 Piecewise-defined Functions Example Consider the function f defined by x if x < 0 f (x) = x if 0 x < Find the value of () f ( ) = ( ) = () f ( ) = = (3) f (.5) = 4.5 =.5 and sketch the graph of f. 4 x if x 6 Solution To draw the graph of f, divide the domain [, 6] into three subintervals: [, 0), [0, ) and [, 6].

7 Piecewise-defined Functions Example Consider the function f defined by x if x < 0 f (x) = x if 0 x < 4 x if x 6 Find the value of () f ( ) = ( ) = () f ( ) = = (3) f (.5) = 4.5 =.5 and sketch the graph of f Solution To draw the graph of f, divide the domain [, 6] into three subintervals: [, 0), [0, ) and [, 6].

8 Piecewise-defined Functions Example Consider the function f defined by x if x < 0 f (x) = x if 0 x < 4 x if x 6 Find the value of () f ( ) = ( ) = () f ( ) = = (3) f (.5) = 4.5 =.5 and sketch the graph of f Solution To draw the graph of f, divide the domain [, 6] into three subintervals: [, 0), [0, ) and [, 6].

9 Piecewise-defined Functions Example Consider the function f defined by x if x < 0 f (x) = x if 0 x < 4 x if x 6 Find the value of () f ( ) = ( ) = () f ( ) = = (3) f (.5) = 4.5 =.5 and sketch the graph of f Solution To draw the graph of f, divide the domain [, 6] into three subintervals: [, 0), [0, ) and [, 6].

10 Definition The absolute value of a real number x, denoted by x, is defined by x if x 0 x = x if x < 0

11 Definition The absolute value of a real number x, denoted by x, is defined by x if x 0 x = x if x < 0 Example () =

12 Definition The absolute value of a real number x, denoted by x, is defined by x if x 0 x = x if x < 0 Example () = () 3 = ( 3)

13 Definition The absolute value of a real number x, denoted by x, is defined by x if x 0 x = x if x < 0 Example () = () 3 = ( 3) = 3

14 Definition The absolute value of a real number x, denoted by x, is defined by x if x 0 x = x if x < 0 Example () = () 3 = ( 3) = 3 a is the distance from a to 0.

15 Definition The absolute value of a real number x, denoted by x, is defined by x if x 0 x = x if x < 0 Example () = () 3 = ( 3) = 3 a is the distance from a to 0. 3units { }} { units 3 { }} { 0 >

16 Definition The absolute value of a real number x, denoted by x, is defined by x if x 0 x = x if x < 0 Example () = () 3 = ( 3) = 3 a is the distance from a to 0. 3units { }} { units 3 0 { }} { > a is always nonnegative

17 a = a 3

18 3 a = a { }} {{ }} { a 0 a >

19 3 a = a { }} {{ }} { a 0 a > a b is the distance from a to b.

20 3 a = a { }} {{ }} { a 0 a > a b is the distance from a to b. Example a = 3, b =

21 3 a = a { }} {{ }} { a 0 a > a b is the distance from a to b. Example a = 3, b = 3

22 3 a = a { }} {{ }} { a 0 a > a b is the distance from a to b. Example a = 3, b = 3 = 5

23 3 a = a { }} {{ }} { a 0 a > a b is the distance from a to b. Example a = 3, b = 3 = 5 = 5 =

24 3 a = a { }} {{ }} { a 0 a > a b is the distance from a to b. Example a = 3, b = 3 = 5 = 5 = distance from 3 to

25 3 a = a { }} {{ }} { a 0 a > a b is the distance from a to b. Example a = 3, b = 3 = 5 = 5 = distance from 3 to a = a

26 3 a = a { }} {{ }} { a 0 a > a b is the distance from a to b. Example a = 3, b = 3 = 5 = 5 = distance from 3 to a = a Example 3 = 9 = 3 =

27 3 a = a { }} {{ }} { a 0 a > a b is the distance from a to b. Example a = 3, b = 3 = 5 = 5 = distance from 3 to a = a Example 3 = 9 = 3 = 3

28 3 a = a { }} {{ }} { a 0 a > a b is the distance from a to b. Example a = 3, b = 3 = 5 = 5 = distance from 3 to a = a Example 3 = 9 = 3 = 3 ( 4) = 6 = 4 =

29 3 a = a { }} {{ }} { a 0 a > a b is the distance from a to b. Example a = 3, b = 3 = 5 = 5 = distance from 3 to a = a Example 3 = 9 = 3 = 3 ( 4) = 6 = 4 = 4

30 Graph of the absolute value function f (x) = x 4

31 4 Graph of the absolute value function f (x) = x Divide the real number line into two subintervals [0, ) and (, 0) f (x) = x if x 0 x if x < 0

32 4 Graph of the absolute value function f (x) = x Divide the real number line into two subintervals [0, ) and (, 0) f (x) = x if x 0 x if x <

33 4 Graph of the absolute value function f (x) = x Divide the real number line into two subintervals [0, ) and (, 0) f (x) = x if x 0 x if x <

34 Example Graph of y = x 5

35 5 Example Graph of y = x y = x if x 0 ( x) if x < 0

36 5 Example Graph of y = x y = x if x ( x) if x <

37 5 Example Graph of y = x y = x if x ( x) if x <

38 5 Example Graph of y = x y = x if x ( x) if x < y = x

39 5 Example Graph of y = x y = x if x ( x) if x < y = x y = x

40 5 Example Graph of y = x y = x if x ( x) if x < y = x y = x y = x +

41 Example Graph y = x 6

42 6 Example Graph y = x y = x if x 0 (x ) if x < 0

43 6 Example Graph y = x y = x if x 0 (x ) if x <

44 6 Example Graph y = x y = x if x 0 (x ) if x <

45 6 Example Graph y = x y = x if x 0 (x ) if x <

46 6 Example Graph y = x y = x if x 0 (x ) if x <

47 6 Example Graph y = x y = x if x 0 (x ) if x <

48 Square-root Function f (x) = x 7 Graph of y = x?

49 Square-root Function f (x) = x 7 Graph of y = x? First, square both sides to get y = x.

50 Square-root Function f (x) = x 7 Graph of y = x? First, square both sides to get y = x. 4 y = x

51 Square-root Function f (x) = x 7 Graph of y = x? First, square both sides to get y = x. Its graph is a parabola. 4 3 y = x x = y

52 Square-root Function f (x) = x 7 Graph of y = x? First, square both sides to get y = x. Its graph is a parabola. Squarring introduces extra points. 4 3 y = x x = y

53 Square-root Function f (x) = x 7 Graph of y = x? First, square both sides to get y = x. Its graph is a parabola. Squarring introduces extra points. x is always non-negative. 4 3 y = x x = y

54 Square-root Function f (x) = x 7 Graph of y = x? First, square both sides to get y = x. Its graph is a parabola. Squarring introduces extra points. x is always non-negative. Graph of y = x is the upper half of the parabola. 4 3 y = x y = x x = y

55 Square-root Function f (x) = x 7 Graph of y = x? First, square both sides to get y = x. Its graph is a parabola. Squarring introduces extra points. x is always non-negative. Graph of y = x is the upper half of the parabola. 4 3 y = x y = x Note y = x y = x, y x = y

56 Square-root Function f (x) = x 7 Graph of y = x? First, square both sides to get y = x. Its graph is a parabola. Squarring introduces extra points. x is always non-negative. Graph of y = x is the upper half of the parabola. 4 3 y = x y = x Note y = x y = x, y Reason a = b = a = b - x = y

57 Square-root Function f (x) = x 7 Graph of y = x? First, square both sides to get y = x. Its graph is a parabola. Squarring introduces extra points. x is always non-negative. Graph of y = x is the upper half of the parabola. 4 3 y = x y = x Note y = x y = x, y Reason a = b = a = b a = b and a, b 0 = a = b - x = y

58 8 Example Sketch the graph of the following () y = x Solution

59 8 Example Sketch the graph of the following () y = x Solution The graph is obtained by moving that of y = x two units down.

60 8 Example Sketch the graph of the following () y = x Solution The graph is obtained by moving that of y = x two units down y = x

61 8 Example Sketch the graph of the following () y = x Solution The graph is obtained by moving that of y = x two units down y = x

62 8 Example Sketch the graph of the following () y = x Solution The graph is obtained by moving that of y = x two units down y = x y = x

63 9 Example Sketch the graph of the following () y = x Solution

64 9 Example Sketch the graph of the following () y = x Solution The graph is obtained by moving that of y = x two units to the right.

65 9 Example Sketch the graph of the following () y = x Solution The graph is obtained by moving that of y = x two units to the right. 4 3 y = x 4 9 6

66 9 Example Sketch the graph of the following () y = x Solution The graph is obtained by moving that of y = x two units to the right. 4 3 y = x 4 9 6

67 9 Example Sketch the graph of the following () y = x Solution The graph is obtained by moving that of y = x two units to the right. 4 3 y = x y = x 4 9 6

68 9 Example Sketch the graph of the following () y = x Solution The graph is obtained by moving that of y = x two units to the right. 4 3 y = x y = x x is defined for x only

69 Composition of Functions 0 To combine functions.

70 Composition of Functions 0 To combine functions. Eg. we may combine the square function and sine function to get sin(x )

71 Composition of Functions 0 To combine functions. Eg. we may combine the square function and sine function to get sin(x ) x x sin(x )

72 Composition of Functions 0 To combine functions. Eg. we may combine the square function and sine function to get sin(x ) x x sin(x ) Definition Let f and g be functions. The composition of g with f, denoted by g f, is the function defined by (g f )(x) = g ( f (x) )

73 Composition of Functions 0 To combine functions. Eg. we may combine the square function and sine function to get sin(x ) x x sin(x ) Definition Let f and g be functions. The composition of g with f, denoted by g f, is the function defined by (g f )(x) = g ( f (x) ) Remark To consider composition, some conditions are needed.

74 Composition of Functions 0 To combine functions. Eg. we may combine the square function and sine function to get sin(x ) x x sin(x ) Definition Let f and g be functions. The composition of g with f, denoted by g f, is the function defined by (g f )(x) = g ( f (x) ) Example Let f and g be functions given by f (x) = x g(x) = sin x Then g f : R R is Remark To consider composition, some conditions are needed.

75 Composition of Functions 0 To combine functions. Eg. we may combine the square function and sine function to get sin(x ) x x sin(x ) Definition Let f and g be functions. The composition of g with f, denoted by g f, is the function defined by (g f )(x) = g ( f (x) ) Example Let f and g be functions given by f (x) = x g(x) = sin x Then g f : R R is (g f )(x) = g(x ) Remark To consider composition, some conditions are needed.

76 Composition of Functions 0 To combine functions. Eg. we may combine the square function and sine function to get sin(x ) x x sin(x ) Definition Let f and g be functions. The composition of g with f, denoted by g f, is the function defined by (g f )(x) = g ( f (x) ) Example Let f and g be functions given by f (x) = x g(x) = sin x Then g f : R R is (g f )(x) = g(x ) = sin(x ) Remark To consider composition, some conditions are needed.

77 Definition (g f )(x) = g ( f (x) ) Example Let f (x) = x and g(x) = x. Find the following (if defined) () (g f )(9) () ( f g)(6) (3) ( f g)()

78 Definition (g f )(x) = g ( f (x) ) Example Let f (x) = x and g(x) = x. Find the following (if defined) () (g f )(9) = g ( f (9) ) () ( f g)(6) (3) ( f g)()

79 Definition (g f )(x) = g ( f (x) ) Example Let f (x) = x and g(x) = x. Find the following (if defined) () (g f )(9) = g ( f (9) ) = g(3) () ( f g)(6) (3) ( f g)()

80 Definition (g f )(x) = g ( f (x) ) Example Let f (x) = x and g(x) = x. Find the following (if defined) () (g f )(9) = g ( f (9) ) = g(3) = () ( f g)(6) (3) ( f g)()

81 Definition (g f )(x) = g ( f (x) ) Example Let f (x) = x and g(x) = x. Find the following (if defined) () (g f )(9) = g ( f (9) ) = g(3) = () ( f g)(6) = f ( g(6) ) (3) ( f g)()

82 Definition (g f )(x) = g ( f (x) ) Example Let f (x) = x and g(x) = x. Find the following (if defined) () (g f )(9) = g ( f (9) ) = g(3) = () ( f g)(6) = f ( g(6) ) = f (4) (3) ( f g)()

83 Definition (g f )(x) = g ( f (x) ) Example Let f (x) = x and g(x) = x. Find the following (if defined) () (g f )(9) = g ( f (9) ) = g(3) = () ( f g)(6) = f ( g(6) ) = f (4) = (3) ( f g)()

84 Definition (g f )(x) = g ( f (x) ) Example Let f (x) = x and g(x) = x. Find the following (if defined) () (g f )(9) = g ( f (9) ) = g(3) = () ( f g)(6) = f ( g(6) ) = f (4) = (3) ( f g)() = f ( g() )

85 Definition (g f )(x) = g ( f (x) ) Example Let f (x) = x and g(x) = x. Find the following (if defined) () (g f )(9) = g ( f (9) ) = g(3) = () ( f g)(6) = f ( g(6) ) = f (4) = (3) ( f g)() = f ( g() ) = f ( )

86 Definition (g f )(x) = g ( f (x) ) Example Let f (x) = x and g(x) = x. Find the following (if defined) () (g f )(9) = g ( f (9) ) = g(3) = () ( f g)(6) = f ( g(6) ) = f (4) = (3) ( f g)() = f ( g() ) = f ( ) undefined

87 Definition (g f )(x) = g ( f (x) ) Example Let f (x) = x and g(x) = x. Find the following (if defined) () (g f )(9) = g ( f (9) ) = g(3) = () ( f g)(6) = f ( g(6) ) = f (4) = (3) ( f g)() = f ( g() ) = f ( ) undefined Remark In order that ( f g)(x) be defined, need g(x) dom ( f ).

88 Definition (g f )(x) = g ( f (x) ) Example Let f (x) = x and g(x) = x +. Find () ( f g)(x) () (g f )(x)

89 Definition (g f )(x) = g ( f (x) ) Example Let f (x) = x and g(x) = x +. Find () ( f g)(x) () (g f )(x) Solution () ( f g)(x) = f (g(x))

90 Definition (g f )(x) = g ( f (x) ) Example Let f (x) = x and g(x) = x +. Find () ( f g)(x) () (g f )(x) Solution () ( f g)(x) = f (g(x)) = f (x + )

91 Definition (g f )(x) = g ( f (x) ) Example Let f (x) = x and g(x) = x +. Find () ( f g)(x) () (g f )(x) Solution () ( f g)(x) = f (g(x)) = f (x + ) = (x + )

92 Definition (g f )(x) = g ( f (x) ) Example Let f (x) = x and g(x) = x +. Find () ( f g)(x) () (g f )(x) Solution () ( f g)(x) = f (g(x)) = f (x + ) = (x + ) = 4x + 4x +

93 Definition (g f )(x) = g ( f (x) ) Example Let f (x) = x and g(x) = x +. Find () ( f g)(x) () (g f )(x) Solution () ( f g)(x) = f (g(x)) = f (x + ) = (x + ) = 4x + 4x + () (g f )(x) = g( f (x))

94 Definition (g f )(x) = g ( f (x) ) Example Let f (x) = x and g(x) = x +. Find () ( f g)(x) () (g f )(x) Solution () ( f g)(x) = f (g(x)) = f (x + ) = (x + ) = 4x + 4x + () (g f )(x) = g( f (x)) = g(x )

95 Definition (g f )(x) = g ( f (x) ) Example Let f (x) = x and g(x) = x +. Find () ( f g)(x) () (g f )(x) Solution () ( f g)(x) = f (g(x)) = f (x + ) = (x + ) = 4x + 4x + () (g f )(x) = g( f (x)) = g(x ) = x +

96 Definition (g f )(x) = g ( f (x) ) Example Let f (x) = x and g(x) = x +. Find () ( f g)(x) () (g f )(x) Solution () ( f g)(x) = f (g(x)) = f (x + ) = (x + ) = 4x + 4x + () (g f )(x) = g( f (x)) = g(x ) = x + Note f g g f in general.

97 3 Exercise Let f : R R and g : R R be functions given by Find () ( f g)(x) () (g f )(x) f (x) = x +, g(x) = x.

98 3 Exercise Let f : R R and g : R R be functions given by Find () ( f g)(x) = x () (g f )(x) = x f (x) = x +, g(x) = x.

99 3 Exercise Let f : R R and g : R R be functions given by Find () ( f g)(x) = x () (g f )(x) = x f (x) = x +, g(x) = x. Definition A function ϕ from a set X into itself satisfying ϕ(x) = x for all x X is called the identity function on X.

100 3 Exercise Let f : R R and g : R R be functions given by Find () ( f g)(x) = x () (g f )(x) = x f (x) = x +, g(x) = x. Definition A function ϕ from a set X into itself satisfying ϕ(x) = x for all x X is called the identity function on X. Remark functions on R. The above example means that ( f g) and (g f ) are the identity

101 3 Exercise Let f : R R and g : R R be functions given by Find () ( f g)(x) = x () (g f )(x) = x f (x) = x +, g(x) = x. Definition A function ϕ from a set X into itself satisfying ϕ(x) = x for all x X is called the identity function on X. Remark functions on R. The above example means that ( f g) and (g f ) are the identity Question Given a function f : X Y, when can we find a function g : Y X such that (g f )(x) = x for all x X?

102 4 Example Let f : X Y be represented by A B C D E Is there any g : Y X satisfying (g f )(x) = x for all x X?

103 4 Example Let f : X Y be represented by A B C D E Is there any g : Y X satisfying (g f )(x) = x for all x X? Solution Yes. Define g() = A

104 4 Example Let f : X Y be represented by A B C D E Is there any g : Y X satisfying (g f )(x) = x for all x X? Solution Yes. Define g() = A Then (g f )(A) = A

105 4 Example Let f : X Y be represented by A B C D E Is there any g : Y X satisfying (g f )(x) = x for all x X? Solution Yes. Define g() = A g() = C Then (g f )(A) = A

106 4 Example Let f : X Y be represented by A B C D E Is there any g : Y X satisfying (g f )(x) = x for all x X? Solution Yes. Define g() = A g() = C Then (g f )(A) = A (g f )(C) = C

107 4 Example Let f : X Y be represented by A B C D E Is there any g : Y X satisfying (g f )(x) = x for all x X? Solution Yes. Define g() = A g() = C g(3) = D Then (g f )(A) = A (g f )(C) = C

108 4 Example Let f : X Y be represented by A B C D E Is there any g : Y X satisfying (g f )(x) = x for all x X? Solution Yes. Define g() = A g() = C g(3) = D Then (g f )(A) = A (g f )(C) = C (g f )(D) = D

109 4 Example Let f : X Y be represented by A B C D E Is there any g : Y X satisfying (g f )(x) = x for all x X? Solution Yes. Define g() = A g() = C g(3) = D g(4) = A Then (g f )(A) = A (g f )(C) = C (g f )(D) = D

110 4 Example Let f : X Y be represented by A B C D E Is there any g : Y X satisfying (g f )(x) = x for all x X? Solution Yes. Define g() = A g() = C g(3) = D g(4) = A g(5) = B Then (g f )(A) = A (g f )(C) = C (g f )(D) = D

111 4 Example Let f : X Y be represented by A B C D E Is there any g : Y X satisfying (g f )(x) = x for all x X? Solution Yes. Define g() = A g() = C g(3) = D g(4) = A g(5) = B Then (g f )(A) = A (g f )(B) = B (g f )(C) = C (g f )(D) = D

112 4 Example Let f : X Y be represented by A B C D E Is there any g : Y X satisfying (g f )(x) = x for all x X? Solution Yes. Define g() = A g() = C g(3) = D g(4) = A g(5) = B g(6) = A Then (g f )(A) = A (g f )(B) = B (g f )(C) = C (g f )(D) = D

113 4 Example Let f : X Y be represented by A B C D E Is there any g : Y X satisfying (g f )(x) = x for all x X? Solution Yes. Define g() = A g() = C g(3) = D g(4) = A g(5) = B g(6) = A g(7) = E Then (g f )(A) = A (g f )(B) = B (g f )(C) = C (g f )(D) = D

114 4 Example Let f : X Y be represented by A B C D E Is there any g : Y X satisfying (g f )(x) = x for all x X? Solution Yes. Define g() = A g() = C g(3) = D g(4) = A g(5) = B g(6) = A g(7) = E Then (g f )(A) = A (g f )(B) = B (g f )(C) = C (g f )(D) = D (g f )(E) = E

115 5 Example Let f : X Y be represented by A B C D E Is there any g : Y X satisfying (g f )(x) = x for all x X? Solution

116 5 Example Let f : X Y be represented by A B C D E Is there any g : Y X satisfying (g f )(x) = x for all x X? Solution Need (g f )(A) = A

117 5 Example Let f : X Y be represented by A B C D E Is there any g : Y X satisfying (g f )(x) = x for all x X? Solution Need (g f )(A) = A (g f )(B) = B

118 5 Example Let f : X Y be represented by A B C D E Is there any g : Y X satisfying (g f )(x) = x for all x X? Solution Need (g f )(A) = A (g f )(B) = B (g f )(C) = C

119 5 Example Let f : X Y be represented by A B C D E Is there any g : Y X satisfying (g f )(x) = x for all x X? Solution Need (g f )(A) = A (g f )(B) = B (g f )(C) = C (g f )(D) = D

120 5 Example Let f : X Y be represented by A B C D E Is there any g : Y X satisfying (g f )(x) = x for all x X? Solution Need (g f )(A) = A (g f )(B) = B (g f )(C) = C (g f )(D) = D (g f )(E) = E

121 5 Example Let f : X Y be represented by A B C D E Is there any g : Y X satisfying (g f )(x) = x for all x X? Solution Need (g f )(A) = A (g f )(B) = B (g f )(C) = C (g f )(D) = D (g f )(E) = E In particular, g() = A

122 5 Example Let f : X Y be represented by A B C D E Is there any g : Y X satisfying (g f )(x) = x for all x X? Solution Need (g f )(A) = A (g f )(B) = B (g f )(C) = C (g f )(D) = D (g f )(E) = E In particular, g() = A g() = C

123 5 Example Let f : X Y be represented by A B C D E Is there any g : Y X satisfying (g f )(x) = x for all x X? Solution Need (g f )(A) = A (g f )(B) = B (g f )(C) = C (g f )(D) = D (g f )(E) = E In particular, g() = A g() = C No such function g.

124 6 Definition A function f : X Y is said to be injective if x x = f (x ) f (x )

125 6 Definition A function f : X Y is said to be injective if Example f (x) = x is injective x x = f (x ) f (x )

126 6 Definition A function f : X Y is said to be injective if Example f (x) = x is injective x x = f (x ) f (x ) 4 3

127 6 Definition A function f : X Y is said to be injective if Example f (x) = x is injective x x = f (x ) f (x ) 4 3 x x = x x

128 6 Definition A function f : X Y is said to be injective if Example f (x) = x is injective x x = f (x ) f (x ) g(x) = x is NOT injective 4 3 x x = x x

129 6 Definition A function f : X Y is said to be injective if Example f (x) = x is injective x x = f (x ) f (x ) g(x) = x is NOT injective x x = x x

130 6 Definition A function f : X Y is said to be injective if Example f (x) = x is injective x x = f (x ) f (x ) g(x) = x is NOT injective x x = x x but ( ) =

131 6 Definition A function f : X Y is said to be injective if Example f (x) = x is injective x x = f (x ) f (x ) g(x) = x is NOT injective x x = x x but ( ) = Remark Geometrically, f is injective means that graph of f intersects every horizontal line in at most one point.

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