Question

Given examples of two functions f: N → N and g: N → N such that  of is onto but f is not onto.

(Hint: Consider f(x) = x + 1 and g(x)= {x−1,if x>1 1, if x=1

Answer 1

Define f: N → N by f(x) = x + 1

and g: N → N by g(x)={x−1,if x>1   1, if x=1

We first show that is not onto. 

For this, consider element 1 in co-domain N. It is clear that this element is not an image of any of the elements in domain N.
∴ f is not onto.