Function families from lattices
As pointed out by Chris Peikert in his comment, the actual definition must be

$f_\mathbf{B}(\mathbf{x}) = \mathbf{x} \mod \mathbf{B} = \mathbf{x} - \mathbf{B} \lfloor\mathbf{B}^{-1} \mathbf{x} \rfloor$.

we can see that the codomain of such $f_\mathbf{B}$ is the fundamental region $\mathcal{P}(\mathbf{B})$ as follows:

Write any given $\mathbf{x}$ as a linear combination of the columns $\mathbf{b}_1$, ..., $\mathbf{b}_n$ of $\mathbf{B}$, that is:

Let $\mathbf{x} = \mathbf{B}\mathbf{z}$ for some $\mathbf{z} \in \mathbb{R}^n$ (it's possible because $\mathbf{B}$ is fullrank, so its columns  span $\mathbb{R}^{n}$).

Let $\mathbf{t} = \mathbf{z} - \lfloor \mathbf{z} \rfloor$, so $\mathbf{t} \in [-1, 1)^n$ and $\mathbf{x} = \mathbf{B}(\mathbf{t} + \lfloor\mathbf{z} \rfloor)$.

Thus,

$f_\mathbf{B}(\mathbf{x}) = \mathbf{x} - \mathbf{B} \lfloor\mathbf{B}^{-1} \mathbf{B}(\mathbf{t} + \lfloor\mathbf{z} \rfloor) \rfloor = \mathbf{x} - \mathbf{B} \lfloor \mathbf{t} + \lfloor  \mathbf{z} \rfloor \rfloor = \mathbf{x} - \mathbf{B} \lfloor  \mathbf{z} \rfloor = \mathbf{B}\mathbf{t}$

and $\mathbf{B}\mathbf{t} \in \mathcal{P}(\mathbf{B})$ by definition of $\mathcal{P}(\mathbf{B})$.