Determine the fixed points (if any).
T(x,y)=(x,2)
This does not belong here. But you are looking for the pairs $\displaystyle (x,y)$ such that $\displaystyle T(x,y)=(x,y)$, i.e. $\displaystyle (x,2)=(x,y)$. Thus we must have $\displaystyle y=2$ and there is no restriction on $\displaystyle x$ (i.e. $\displaystyle x\in \mathbb{R}$).