I'm pretty stuck on the following question
$f$ on $\mathbb{R}$ given by $xfy\Leftrightarrow (y(2x-3)-3x=y(x^2-2x)-5x^3)$ is a function.
Let $g$ be the restriction of $f$ to $\mathbb{Z}^+$, implying $g(n) = f(n),\,n \in\mathbb{Z}^+$
Determine $a\in R\,$, so $g\in \Theta(n^a)$
Let's express $y=f(x)$: $$(x^2-2x-2x+3)y=5x^3-3x \implies y =\frac{5x^3-3x}{x^2-4x+3} $$ So, $a=3-2=1$ by the leading exponents.
Update: Formally, you have to prove that there are constants $A,B>0$ such that $An\le f(n)\le Bn\ $ (for $n\ge n_0$).