Solving a functional relation $f\left( x \cdot f(y)\right)=x^2 \cdot y^a$ I have this functional relation - $$f\left( x \cdot f(y)\right)=x^2 \cdot y^a$$ which I am trying to solve. 
I put $x=1$, then I put $f(y)=\dfrac{1}{x}$. I also tried out $y=f^{-1}(1)$, but it doesn't seem to work out. Please help me out. Thank you.

For $x>0$:
$$f(f(x))=x^a$$
$$f(\sqrt x f(1))=x$$
so
$$f(x)=f(f(\sqrt xf(1)))=\sqrt{x^a}f(1)^a$$
This would determine $f$, chosen $f(1)$ and $a$.
Now, let's see if we really can choose:
$$x^2y^a=f(xf(y))=f(x\sqrt{y^a}f(1)^a)=x^{a/2}y^{a^2/4}f(1)^{3a/2}$$
For $y=1$:
$$x^4=x^af(1)^{3a}\quad\forall x>0$$
hence $x^{4-a}$ is constant, so it must be $a=4$, $f(1)=1$. So finally
$$f(x)=x^2$$
