 Solution

A rational function is a quotient $r=P/Q$ of polynomials with $Q\ne 0$. Cancel common factors. It determines a map on the Riemann sphere by assigning infinity at poles and assigning at infinity the limit in the sphere. Equivalently homogenize $P,Q$ to the same degree $d=\max (\mathrm{deg}P,\mathrm{deg}Q)$ and set $f([Z:W])=[P_h(Z,W):Q_h(Z,W)]$. They have no common projective zero: finite ones would contradict coprimality, and at infinity at least one has a nonzero leading coefficient. This also proves the map is holomorphic.

The degree of the nonconstant rational function is $d$. The degree of a holomorphic map of compact Riemann surfaces is the number of preimages of any target point, counted with local mapping multiplicity. For target $[a:b]$, the homogeneous equation $b{P}_h-a{Q}_h=0$ is a nonzero homogeneous polynomial of degree $d$. Over $\mathbb{C}$ it factors into $d$ linear factors, with multiplicity, giving exactly $d$ projective zeros. Locally its zero order is precisely the map's multiplicity, since the other target-coordinate denominator is nonzero. The two degrees are equal.

There is a Möbius transformation sending any ordered triple of distinct points to $0,1,\infty$. When all are finite one can use $T_A(z)=(z-a_1)(a_2-a_3)/[(z-a_3)(a_2-a_1)]$; the formulas for infinity are its limits. Define $T_B$ in the same way. Then $T_B^{-1}\circ T_A$ sends $A$ to $B$ and restricts to a biholomorphism $X\setminus A\to X\setminus B$.