check the picture below.
notice, the hollow dot endpoints, is where the subfunction does not include that value of "x", and where the dot is a solid dot, that value is included in the subfunction.
so the piece-wise is simply the composition of all its subfunctions, in this case 3 subfunctions, within their range, notice the lines continue on the dashed line part, but that part is not included in the piece-wise because is outside the range provided for "x".
