<s>[INST] $f(x)=?$ if we have $f\left(\frac{x}{x^2+x+1}\right)=\frac{x}{x^2-x+1}$ $f(x)=?$
If we have $$f\left(\frac{x}{x^2+x+1}\right)=\frac{x}{x^2-x+1}$$ to fractions are very similar. I don't have an idea to find $f(x)$. Can someone show me a clue ?
 [/INST] Let $y = \frac x{x^2+x+1}$ and $z = \frac x{x^2-x+1}$. Then we have
\begin{align}
y(x^2+x+1) = x &\implies y(x^2-x+1)=x -2xy\\
&\implies y=z(1-2y)\\
&\implies z=\frac y{1-2y}\\
\end{align}
which gives us $f(y) = \frac{y}{1-2y}$.
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