BALTIC-WAY 2007 Problem 5

A function $f$ is defined on the set of all real numbers except $0$ and takes all real values except $1$ . It is also known that $\color{white}\ . \ \color{black}\ f(xy)=f(x)f(-y)-f(x)+f(y)$ for any $x,y\not= 0$ and that $\color{white}\ . \ \color{black}\ f(f(x))=\frac{1}{f(\frac{1}{x})}$ for any $x\not\in\{ 0,1\}$ . Determine all such functions $f$ .