Let $F = \{f\mid f\colon \mathbb R \to \mathbb R\}$, and define a relation $S$ on $F$ as follows: $S = \{(f,g) ∈ F \times F \mid \exists h \in F :f = h\circ g\}$. Let $f$, $g$ and $h$ be the functions from $\mathbb R$ to $\mathbb R$ defined by the formulas $f(x) = x^2 + 1$, $g(x) = x^3 + 1$, and $h(x) = x^4 + 1$. Prove that $h\,S\,f$, but that it is not the case that $g\, S\, f$.
By letting $j(x) = x^2 - 2x + 2$, then $h = j \circ f$, thus $h\, S\,f$.
I've been struggling to show that it is not the case that $g\,S\,f$. In specific, I need to show that $g$ is not equal to $j \circ f$, for arbitrary $j \colon \mathbb R \to \mathbb R$.
Any help would be appreciated.