How can I use the Implicit Function Theorem without hypothesis about the Jacobian be invertible

Question

Let $f,g: \mathbb{R}^n \to \mathbb{R}$ such that $g(x) = f(x) + (f(x))^5$. If $g \in C^r$ then $f \in C^r$.

This question is of my practice list of Implicit Function Theorem, but I cannot see how I use the IFT in this question. Can someone help me? I dont want a complete solution, but, at least, a hint about how apply the IFT to this question.

Answer 1

HINT: Consider the function $\phi\colon\Bbb R\to\Bbb R$, $\phi(t)=t+t^5$. Can you express $g$ in terms of $\phi$ and $f$? What do you need to express $f$ in terms of $\phi$ and $g$?