Question

Give examples of two functions $f:N\to N$ and $g:N\to N$ such that $g\circ f$ is onto, but $f$ is not onto.

Answer 1

If $f\left(x\right)=x+1$ ang $g\left(x\right)=\{\begin{array}{c}x-1,ifx>1\\ 1,ifx=1\end{array}$, then $f:N\to N$ is not onto.
Range $\left(f\right)=N-\left\{1\right\}\ne$ Co-domain of $f$
Now, $g\circ f\left(x\right)=g\left(f\right(x\left)\right)=g(x+1)=x+1-1=x$.
Clearly, $g\circ f$, being identity funciton, is onto.