Conditional equations for monomial functions

In the first part of this paper, we consider monomial functions $f$ of degree $n\in\mathbb{N}$ that satisfy the additional equation $y^nf(x)=x^nf(y)$ under the condition $y=a_mx^m+a_{m-1}x^{m-1}+\cdots+a_1x+a_0$, $m\in\mathbb{N}$, with $a_i\in\mathbb{R}$, $i=0,\dots,m$ and $a_m\neq 0$, $a_0\neq 0$. We prove that $f(x)=x^nf(1)$ for all $x\in\mathbb{R}$. In the second part, we consider monomial functions $f$ of degree $3$ that satisfy the additional equation $f\left(x^m\right)=x^{3(m-1)}f(x)$ with $|m|\geqslant 2$, $m\in\mathbb{Z}$. We prove that $f(x)=x^3f(1)$ for all $x\in\mathbb{R}$. Counterexamples are presented for the case $m=-1$ in a general context.