proving onto function of composite functions.

Question

Let $X, Y, Z$ be arbitrary sets. Suppose $\alpha$ is a function from $X$ to $Y$ and $\beta$ is a function from $Y$ to $Z$ such that $\beta\circ\alpha$ is an onto function. How do I prove that $\beta$ is an onto function? I always get confused when it's proving.

Answer 1

For any $z\in Z$ there exist $x\in X$ such that $(\beta\circ\alpha)(x)=z$, since $\beta\circ\alpha$ is onto. Now let $y:=\alpha(x)$ we have $\beta(y)=z$ with $y\in Y$, hence $\beta $ is onto.