Functional equation substitution

Question

For functional equations like $f(f(n)) + f^2(n) = n^2 + 3n + 3 $, can you make substitutions like $f(n) = g(n) + kn + c$, where $g(n)$ has no constant terms or any terms of $n$ ( if $g(n)$ itself does not contain any constant terms or terms of $n$, we can set $k = 0$ and $c = 0$)

Answer 1

I am not entirely sure what you mean.
Suppose $$ f(f(n)) + f(n)^2 = n^2 + 3n + 3 $$ for all $n$. Substutute $$ f(n) = g(n) + kn + c \tag{1}$$ to get $$ g \left( g \left( n \right) +kn+c \right) +kg \left( n \right) +{k}^{2 }n+kc+c+ \left( g \left( n \right) \right) ^{2}+2\,g \left( n \right) kn+2\,g \left( n \right) c+{k}^{2}{n}^{2}+2\,knc+{c}^{2}={n}^ {2}+3\,n+3 $$ We can do this if we know the functions $f$ and $g$ are related by $(1)$. Or, if we do not already have a $g$ to use, we can rearrange $(1)$ as $$ g(n) = f(n) - kn - c \tag{2}$$ and take that as the definition of the function $g$.