Look at the example 2.1.7 at page 19 of these notes (which is the same example present in Mumford's red book at page 21). The author shows that a regular function isn't a ratio of two polynomial, and in particular in the above example inside the open set $U$ we have the following regular function: $\phi(p)=\left\{\begin{array} {lll} \frac{X_1}{X_2} & \textrm{if $p\in X_2\neq 0$}\\\\ \frac{X_3}{X_4} & \textrm{if $p\in X_4\neq 0$} \end{array}\right.$
I agree with the fact that $\frac{X_1}{X_2}=\frac{X_3}{X_4}$ in $K(X)$ when $X_2\neq0$ and $X_4\neq0$, but the point is that such two functions shoud be the same in the whole $V$ to make sensible the example. Infact in in the intersection $\{X_2\neq0\}\cap\{X_4\neq0\}$ we have the regular function described by $\frac{X_1}{X_2}$ or $\frac{X_3}{X_4}$, but what about the rest of $U$?