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 $$f\big(yf(x)\big)=x^2y^4$$
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 $$f\big(yf(x)\big)=x^2y^4$$
$ 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 $f(yf(x))=x^2y^4$
A bit late to the party, but:
In the case $y=1$, we get $f(f(x)) = x^2$, so $f(x)$, it it exists at all, must equal $x^\sqrt{2}$ for all possible $x$
But now we run into trouble as $f(x):=x^{\sqrt{2}}$ does not satisfy that property for y $\neq 1$. Indeed, $f(yf(x)) = y^\sqrt{2} x^2$