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


 Emil Singleton
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1 Piecewisedefined Functions Example Consider the function f defined by x if x < 0 f (x) = x if 0 x < 4 x if x 6
2 Piecewisedefined 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 Piecewisedefined 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 Piecewisedefined 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 Piecewisedefined 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 Piecewisedefined 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 Piecewisedefined 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 Piecewisedefined 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 Piecewisedefined 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 Squareroot Function f (x) = x 7 Graph of y = x?
49 Squareroot Function f (x) = x 7 Graph of y = x? First, square both sides to get y = x.
50 Squareroot Function f (x) = x 7 Graph of y = x? First, square both sides to get y = x. 4 y = x
51 Squareroot 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 Squareroot 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 Squareroot 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 nonnegative. 4 3 y = x x = y
54 Squareroot 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 nonnegative. Graph of y = x is the upper half of the parabola. 4 3 y = x y = x x = y
55 Squareroot 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 nonnegative. 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 Squareroot 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 nonnegative. 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 Squareroot 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 nonnegative. 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.
You know from calculus that functions play a fundamental role in mathematics.
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