Question

Consider f : {1, 2, 3} → { a , b , c } given by f (1) = a , f (2) = b and f (3) = c . Find f −1 and show that ( f −1 ) −1 = f .

Answer 1

The provided function is f:{ 1,2,3 }→{ a,b,c } is given by, f( 1 )=a , f( 2 )=b and f( 3 )=c .

Consider the inverse of the function g:{ a,b,c }→{ 1,2,3 } as g( a )=1 , g( b )=2 and g( c )=3 .

fog( a )=f( g( a ) ) =f( 1 ) =a

fog( b )=f( g( b ) ) =f( 2 ) =b

fog( c )=f( g( c ) ) =f( 3 ) =c

gof( 1 )=g( f( 1 ) ) =g( a ) =1

gof( 2 )=g( f( 2 ) ) =g( b ) =2

gof( 3 )=g( f( 3 ) ) =g( c ) =3

The range of the function gof and fog is given by,

gof= I x →X={ 1,2,3 } fog= I y →Y={ a,b,c }

Thus, the inverse of the function f exists and f −1 =g .

The provided function is f −1 :{ a,b,c }→{ 1,2,3 } is given by, f −1 ( a )=1 , f −1 ( b )=2 and f −1 ( c )=3 . Consider the inverse of the function h:{ 1,2,3 }→{ a,b,c } as h( 1 )=a , h( 2 )=b and h( 3 )=c .

hog( a )=h( g( a ) ) =h( 1 ) =a

hog( b )=h( g( b ) ) =h( 2 ) =b

hog( c )=h( g( c ) ) =h( 3 ) =c

goh( 1 )=g( h( 1 ) ) =g( a ) =1

goh( 2 )=g( h( 2 ) ) =g( b ) =2

goh( 3 )=g( h( 3 ) ) =g( c ) =3

The range of the function gof and fog is given by,

goh= I x →X={ 1,2,3 } hog= I y →Y={ a,b,c }

Thus, the inverse of the function f −1 exists and ( f −1 ) −1 =h .

The value of h is equal to that of the function f .

Thus, the value of ( f −1 ) −1 is f .