Question

Give an example of two functions $f:N\to N$ and $g:N\to N$ such that gof is onto but f is not onto.

Answer 1

Define $f:N\to Nbyf\left(x\right)=x+1$
and $g:N\to N$ by $g\left(x\right)=\{\begin{array}{c}x-1,ifx>1\\ 1,ifx=1\end{array}$
We first show that f is not onto
For this, consider element 1 in co-domain N.
Let $f\left(x\right)=1\Rightarrow x+1=1\Rightarrow x=0$, which is not a natural number.
Therefore, f is not onto. Now, gof: $N\to N$ is defined by,
$gof\left(x\right)=g\left(f\right(x\left)\right)=g(x+1)=(x+1)-1[\because x\in N\Rightarrow (x+1)>1]$
Then, it is clear that for y in N, there exists $x=y\in N$ such that gof(x)=y.
Hence, gof is onto.