Condition on Vector Boolean Function to be Bijective Suppose the vector boolean function be 

$$\begin{align}
f:F^n_2 \longrightarrow F_2^n \\
(x_1,\dots ,x_n) \longrightarrow (x_2,\dots x_n,g) \\  \\
g:F^n_2 \longrightarrow F_2 \\
(x_1,\dots ,x_n) \longrightarrow \{0,1\}
\end{align}$$

What should be the condition on the Boolean function $g$ so that $f$ becomes bijective i.e. 1-1 and Onto?
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 $x_1 = y_1$, $x_2 = y_2$, ..., $x_n = y_n$