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
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
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
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
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
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].
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. 4 3 4 6 Solution To draw the graph of f, divide the domain [, 6] into three subintervals: [, 0), [0, ) and [, 6].
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. 4 3 4 6 Solution To draw the graph of f, divide the domain [, 6] into three subintervals: [, 0), [0, ) and [, 6].
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. 4 3 4 6 Solution To draw the graph of f, divide the domain [, 6] into three subintervals: [, 0), [0, ) and [, 6].
Definition The absolute value of a real number x, denoted by x, is defined by x if x 0 x = x if x < 0
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 () =
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)
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
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.
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 >
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
a = a 3
3 a = a { }} {{ }} { a 0 a >
3 a = a { }} {{ }} { a 0 a > a b is the distance from a to b.
3 a = a { }} {{ }} { a 0 a > a b is the distance from a to b. Example a = 3, b =
3 a = a { }} {{ }} { a 0 a > a b is the distance from a to b. Example a = 3, b = 3
3 a = a { }} {{ }} { a 0 a > a b is the distance from a to b. Example a = 3, b = 3 = 5
3 a = a { }} {{ }} { a 0 a > a b is the distance from a to b. Example a = 3, b = 3 = 5 = 5 =
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
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
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 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
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 =
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
Graph of the absolute value function f (x) = x 4
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
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 < 0.5 0.5 - -
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 < 0.5 0.5 - -
Example Graph of y = x 5
5 Example Graph of y = x y = x if x 0 ( x) if x < 0
5 Example Graph of y = x y = x if x 0 0.5 ( x) if x < 0 - - -0.5 -
5 Example Graph of y = x y = x if x 0 0.5 ( x) if x < 0 - - -0.5 -
5 Example Graph of y = x y = x if x 0 0.5 ( x) if x < 0 - - -0.5 - y = x
5 Example Graph of y = x y = x if x 0 0.5 ( x) if x < 0 - - -0.5 - y = x y = x
5 Example Graph of y = x y = x if x 0 0.5 ( x) if x < 0 - - -0.5 - y = x y = x y = x +
Example Graph y = x 6
6 Example Graph y = x y = x if x 0 (x ) if x < 0
6 Example Graph y = x y = x if x 0 (x ) if x < 0.5 0.5-3
6 Example Graph y = x y = x if x 0 (x ) if x < 0.5 0.5-3
6 Example Graph y = x y = x if x 0 (x ) if x < 0.5 0.5-3.5 0.5 - - 3
6 Example Graph y = x y = x if x 0 (x ) if x < 0.5 0.5-3.5 0.5 - - 3
6 Example Graph y = x y = x if x 0 (x ) if x < 0.5 0.5-3.5 0.5 - - 3
Square-root Function f (x) = x 7 Graph of y = x?
Square-root Function f (x) = x 7 Graph of y = x? First, square both sides to get y = x.
Square-root Function f (x) = x 7 Graph of y = x? First, square both sides to get y = x. 4 y = x 3 - - 3 4 - -
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 - - 3 4 - - x = y
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 - - 3 4 - - x = y
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 - - 3 4 - - x = y
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 - - 3 4 - - x = y
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 0 - - 3 4 - - x = y
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 0 - - 3 4 - Reason a = b = a = b - x = y
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 0 - - 3 4 - Reason a = b = a = b a = b and a, b 0 = a = b - x = y
8 Example Sketch the graph of the following () y = x Solution
8 Example Sketch the graph of the following () y = x Solution The graph is obtained by moving that of y = x two units down.
8 Example Sketch the graph of the following () y = x Solution The graph is obtained by moving that of y = x two units down. 4 3 - - y = x.5 5 7.5 0.5 5
8 Example Sketch the graph of the following () y = x Solution The graph is obtained by moving that of y = x two units down. 4 3 - - y = x.5 5 7.5 0.5 5
8 Example Sketch the graph of the following () y = x Solution The graph is obtained by moving that of y = x two units down. 4 3 - - y = x y = x.5 5 7.5 0.5 5
9 Example Sketch the graph of the following () y = x Solution
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.
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
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
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
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. 4 9 6
Composition of Functions 0 To combine functions.
Composition of Functions 0 To combine functions. Eg. we may combine the square function and sine function to get sin(x )
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 )
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) )
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.
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.
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.
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.
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)()
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)()
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)()
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)()
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)()
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)()
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)()
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() )
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 ( )
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
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 ).
Definition (g f )(x) = g ( f (x) ) Example Let f (x) = x and g(x) = x +. Find () ( f g)(x) () (g f )(x)
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))
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 + )
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 + )
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 +
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))
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 )
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 +
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.
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.
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.
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.
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
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?
4 Example Let f : X Y be represented by A B C D E 3 4 5 6 7 Is there any g : Y X satisfying (g f )(x) = x for all x X?
4 Example Let f : X Y be represented by A B C D E 3 4 5 6 7 Is there any g : Y X satisfying (g f )(x) = x for all x X? Solution Yes. Define g() = A
4 Example Let f : X Y be represented by A B C D E 3 4 5 6 7 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
4 Example Let f : X Y be represented by A B C D E 3 4 5 6 7 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
4 Example Let f : X Y be represented by A B C D E 3 4 5 6 7 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
4 Example Let f : X Y be represented by A B C D E 3 4 5 6 7 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
4 Example Let f : X Y be represented by A B C D E 3 4 5 6 7 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
4 Example Let f : X Y be represented by A B C D E 3 4 5 6 7 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
4 Example Let f : X Y be represented by A B C D E 3 4 5 6 7 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
4 Example Let f : X Y be represented by A B C D E 3 4 5 6 7 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
4 Example Let f : X Y be represented by A B C D E 3 4 5 6 7 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
4 Example Let f : X Y be represented by A B C D E 3 4 5 6 7 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
4 Example Let f : X Y be represented by A B C D E 3 4 5 6 7 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
5 Example Let f : X Y be represented by A B C D E 3 4 5 6 7 Is there any g : Y X satisfying (g f )(x) = x for all x X? Solution
5 Example Let f : X Y be represented by A B C D E 3 4 5 6 7 Is there any g : Y X satisfying (g f )(x) = x for all x X? Solution Need (g f )(A) = A
5 Example Let f : X Y be represented by A B C D E 3 4 5 6 7 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
5 Example Let f : X Y be represented by A B C D E 3 4 5 6 7 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
5 Example Let f : X Y be represented by A B C D E 3 4 5 6 7 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
5 Example Let f : X Y be represented by A B C D E 3 4 5 6 7 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
5 Example Let f : X Y be represented by A B C D E 3 4 5 6 7 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
5 Example Let f : X Y be represented by A B C D E 3 4 5 6 7 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
5 Example Let f : X Y be represented by A B C D E 3 4 5 6 7 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.
6 Definition A function f : X Y is said to be injective if x x = f (x ) f (x )
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 )
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
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
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
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 4 3 x x = x x
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 4 3 x x = x x but ( ) =
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 4 3 x x = x x but ( ) = Remark Geometrically, f is injective means that graph of f intersects every horizontal line in at most one point.