Condition on Vector Boolean Function to be Bijective
The condition is that:

$$g(0, x_2, x_3, ..., x_n) \ne g(1, x_2, x_3, ..., x_n)$$

for all $x_2, x_3, ..., x_n$

This can easily be derived from the condition that implies bijectivity of $f$; that is,  $f(x_1, x_2, ..., x_n) = f(y_1, y_2, ..., y_n)$ implies that $x_1 = y_1$, $x_2 = y_2$, ..., $x_n = y_n$