Find all functions $f:\mathbb R\to\mathbb R$ such that $f\big(yf(x)\big)=x^2y^4$

Question

It's my last question. Just give me advice how to start.

Find all such functions $$f:\mathbb R\to \mathbb R\text,$$ for all real $x$ and $y$, the equality $$f\big(yf(x)\big)=x^2y^4$$

Answer 1

$ f(yf(1))=y^4, $ so $f(1)\ne0$, so $f(y)=\left(\frac{y}{f(1)}\right)^4$. Now putting $y=1$ in last equation we get $f(1)=1$ and $f(y)=y^4$. Now it is easy too see that $f(y)=y^4$ cannot satisfy the equation $f(yf(x))=x^2y^4$