2. If $f_{1}(x)=\frac{x}{x-1}, f_{2}(x)=\frac{1}{1-x}$ and $f_{n+2}(x)=f_{n+1}\left(f_{n}(x)\right)$ for $n \geq 1$. Calculate $f_{2015}(2015)$.
Solution. Let's apply the given recurrence formula for several functions:

$$
f_{3}(x)=1-x, f_{4}(x)=\frac{x}{x-1}, f_{5}(x)=\frac{x-1}{x}, f_{6}(x)=\frac{1}{x}, f_{7}=f_{1}, f_{8}=f_{2} \text { and so on. }
$$

We observe that there is a periodic repetition, with a period of 6. Therefore,

$$
f_{2015}(x)=f_{5}(x) \text { and } f_{2015}(2015)=\frac{2014}{2015} .
$$