This is from the Exercise 2 of Chapter I, section 3 in Shafaravich's book. The question is that every element in $k(X)$ can be written as $u(x)+v(x)y$ for some rational functions $u(x),v(x) \in k[x,x^{-1}]$.
My attempt is below; Since $y^{2}-x^{2}-x^{3}$ is prime (Use Eisenstein criterion by thinking $k[x]$ as PID and the prime element $x+i$ in $k[x]$ such that $x+i$ cannot divides $x^{2}+x^{3}$ (one can check by constructing the quotient), we know $k[X] = k[x,y]/(y^{2}-x^{2}-x^{3})$, i.e., the coordinate ring is $X$. Thus, any element in $k[X]$ can be written as $f_{1}(x)+f_{2}(x)y$ for some polynomials $f_{1},f_{2} \in k[x]$. Thus any elements in $k(X)$ is of a form $(f_{1}+f_{2}y)/(g_{1}+g_{2}y)$.
However, I'm stuck at this point. How can we change the form $(f_{1}+f_{2}y)/(g_{1}+g_{2}y)$ into $u(x)+v(x)y$? Any hints will be appreciated.