Input voltage is not a truth-table caption; it creates a region and then an output. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
Input one volt leaves the output high
Input 1 V gives base current 0 A and collector current 0 A. The checked region is cutoff, so the output is 12 V and the state is HIGH.
Vin=1V⇒IC=0A,Vout=12V
Input three volts creates an analog middle
Input 3 V gives base current 1/2 A and collector current 3/2 A. The checked region is active, so the output is 6 V and the state is MIDDLE.
Vin=3V⇒IC=23A,Vout=6V
Input five volts saturates the pull-down path
Input 5 V gives base current 1 A and collector current 3 A. The checked region is saturation, so the output is 0 V and the state is LOW.