Question

Give examples of two functions f : N → Z and g : Z → Z such that g o f is injective but g is not injective. (Hint: Consider f ( x ) = x and g ( x ) = )

Answer 1

It is given that the function f( x ) is defined in the domain f:N→Z and the function g( x ) is defined in the domain g:Z→Z.

Consider the functions f( x )=x and g( x )=| x |.

First, show that the function g( x ) is not injective.

Let x=1;x=−1

g( −1 )=| −1 | =1 g( 1 )=| 1 | =1

The value of the function g( x ) is same for the two different values of x. So, the function is not injective because g( 1 )=g( −1 ) but 1≠−1.

Substitute x for f( x ) in gof( x ).

gof( x )=g( f( x ) ) =g( x ) =| x |

Now, show that the function gof( x ) is not injective.

gof| x |=gof| y | | x |=| y | x=y

As the value of gof| x |=gof| y |, so the function gof is injective.

Hence, it is proved that gof is injective, but g is not injective.