Question

State if the given functions are inverses

Answer 1

Two functions f and g are inverse functions if and only if both of their

compositions are the identity functions.

That means [f o g](x) = x and [g o f](x) = x.

1)

g(x)  =  4 - (3x)/2   and  f(x)  =  x/2 + 3/2

f o g (x)  =  f( g(x) )

              =  f( 4 - (3x)/2)

              =  [ 4 - (3x)/2 ]/2 + 3/2

              =  4/2 - (3x)/4 + 3/2

              =  (4 + 3)/2 - (3x)/4

              =  7/2 - (3x)/4

g o f (x)  =  g ( (f(x) )

              =  g(  x/2 + 3/2 )

              =  4 - 3[x/2 + 3/2]/2

              =  4 - 3[x/4 + 3/4]

              =  4 - (3x)/4 - 9/4

              =  (16 - 9)/4 - (3x)/4

              =  7/4 - (3x)/4

Here, f o g (x) and g o f (x) are not equal

Hence, those are not an inverse functions.

 

3)

f(n)  =  (-16 + n) / 4  and g(n)  =  4n + 16

f o g (n)  =  f( g(n) )

              =  f( 4n + 16)

              =  [ - 16 + ( 4n + 16) ] / 4

              =  [ 4n ] / 4

              =  n

g o f (n)  =  g(  f(n) )

              =  g( (-16 + n) / 4 )

              =  4[(-16 + n) / 4 )] + 16

              =  -16 + n + 16

              =  n

Here, f o g (n)  =  g o f (n) = n

Therefore f(n) and g(n) are identitity functions

Hence, f(n and g(n) are inverse functions.

Answer :

1)  f(x) and g(x) are not an inverse functions.

3)  f(n and g(n) are inverse functions.