For any rational map represented by $(\frac{x^4+3y}{x^2+1}, \frac{x+1}{y})$ in affine coordinates, write down the corresponding representation $[F_1(X, Y, Z) : F_2(X, Y, Z) : F_3(X, Y, Z)]$ in projective coordinates.
My understanding is that a point in affine coordinates $(X', Y')$ has a projective representation of $(X', Y', 1)$. So is the answer just $(\frac{x^4+3y}{x^2+1}, \frac{x+1}{y}, 1)$? That doesn't seem right to me. How should I properly convert these points?
Multiply by $y(x^2+1)$ so your map is $((x^4+3y)y, (x+1)(x^2+1), y(x^2+1))$. Now further you want $X,Y,Z$ to be homogeneous so set $x=\frac{X}{Z}$ and $y=\frac{Y}{Z}$ and now multiply by the lowest power of $Z$ to clear the denominators.