Question

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

Answer 1

Let us consider a function f : N → N given by f(x) = x +1 , which is not onto.
[This not onto because if we take 0 in N (co-domain), then,
0=x+1
$\Rightarrow$x=-1$\notin N$]

Let us consider g : N → N given by
$$\begin{gathered} g\left(x\right)=\left\{\begin{array}{l}x-1,\text{if}x>1\\ 1,\text{if}x=1\end{array}\right. \\ \text{Now, let us find}\left(gof\right)\left(x\right) \\ \text{Case 1:}x>1 \\ \left(gof\right)\left(x\right)=g\left(f\left(x\right)\right)=g\left(x+1\right)=x+1-1=x \\ \text{Case 2:}x=1 \\ \left(gof\right)\left(x\right)=g\left(f\left(x\right)\right)=g\left(x+1\right)=1 \\ \text{From case-1 and case-2,}\left(gof\right)\left(x\right)=x,\forall x\in N, \\ \text{which is an identity function and, hence, it is onto.} \end{gathered}$$