Image link at bottom I'm not sure how to go about showing that $y=\phi_1(x)$ gives the curve $4y^3-3y-x=0$ locally. I may be able to show that $DF(a) = D\phi_1(a)$, but that doesn't prove it over the whole domain. How do I approach this? Specifically, what must I show for a clear proof?
https://i.stack.imgur.com/R5sgW.jpg
Edit: It may not have been clear that I meant $a$ to be a single arbitrary point, not all $a ∈ ℝ$. $\phi_1(x)={1\over2}((x+\sqrt{x^2-1})^{1/3}+(x+\sqrt{x^2-1})^{-1/3})$ for $x>1$.