MathDB

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

Problem Statement