Question

Given examples of two functions f : N → N and g : N → N such that g o f is onto but f is not onto. (Hint: Consider f ( x ) = x + 1 and

Answer 1

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

Consider f( x )=x+1 and g( x )={ x−1 if x>1 1 if x=1 }.

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

Consider an element 1 that exists in the co-domain N. Also, this element is not the image of any element in the domain N.

Thus, the function f is not onto.

Now, find the value of gof( x ).

gof( x )=g( f( x ) ) =g( x+1 ) =( x+1 )−1 =x

It can be observed that for y∈N, there exist x=y∈N such that gof( x )=y.

Hence, it is proved that gof is onto, but f is not onto.