To see if it is surjective, simply check if every element $y\in\mathbb Z$ can appear in $A$. This means:
- Take an arbitrary $y\in \mathbb Z$
- Find such an $x\in \mathbb R$ that $(x,y)\in A$.
On the other hand, if you want to prove a function is not surjective, simply find one particular value of $y$ such that $(x,y)$ is not in $A$ for any value $x$.
For injectivity, if you want to prove injectivity, take two pairs $(x_1, y_1)$ and $(x_2, y_2)$ such that $y_1=y_2$. If you can conclude that $x_1=x_2$, then the function is injective.
If you want to prove that the function is not injective, simply find two values of $x_1,x_2$ and one value of $y$ such that $(x_1,y)$ and $(x_2,y)$ are both in $A$.