Question 109
Find all finite and non-empty sets $A$ of functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that:
For all $f_{1}, f_{2} \in A$, there exists $g \in A$ such that for all $x, y \in \mathbb{R}$
$$
f_{1}\left(f_{2}(y)-x\right)+2 x=g(x+y)
$$
For all $f_{1}, f_{2} \in A$, there exists $g \in A$ such that for all $x, y \in \mathbb{R}$
$$
f_{1}\left(f_{2}(y)-x\right)+2 x=g(x+y)
$$