# Solve a system of functional equations

Find all functions $$f:mathbb Rtomathbb R$$ such that for all $$xinmathbb R$$, $$f(xf(x))=f(x)^2$$ & $$f(f(x))=x$$.

Evaluating at zero, $$f(0)=f(0)^2$$ and $$f(f(0))=0$$.
Evaluating at one, $$f(f(1))=f(1)^2$$ and $$f(f(1))=1$$.

So $$f(1)=pm 1$$ and $$f(0)=frac 12pmfrac 12$$. But I can’t find more information.