Question

If $f:A\to B$ and $g:B\to C$ are onto functions show that gof is an onto function.

Answer 1

since $g:B\to C$ is onto

suppose $z\in C$,there exists a pre-image in $B$

Let the pre-image be $y$

Hence $y\in B$ such that $g\left(y\right)=z$

similarly $f:A\to B$ is onto

suppose $y\in B$,there exists a pre-image in $A$

Let the pre-image be $x$

Hence $x\in A$ such that $f\left(x\right)=y$

Now $gof\left(x\right)=g\left(f\right(x\left)\right)=g\left(y\right)=z$

So for every $x$ in $A$,there is an image $z$ in $C$

Thus $gof$ is an onto function