The transcription of inducible genes involves signaling pathways that induce DNA binding of the downstream transcription factors to form functional promoter states. How the transcription dynamics is linked to the temporal variations of activation signals is far to be fully understood. In this work, we develop a mathematical model with multiple promoter states to address this question. Each promoter state has its own activation and inactivation rates, and is selected randomly with a probability that may change in time. Under the activation of constant signals, our analysis shows that if only the activation rates differ among the promoter states, then the mean transcription level $m(t)$ displays only a monotone or monophasic growth pattern. In a sharp contrast, if the inactivation rates change with the promoter states, then $m(t)$ may display multiphasic growth patterns. Upon the activation of signals that oscillate periodically, $m(t)$ also oscillates later, almost periodically in the same frequency, but the magnitude decreases in the frequency and is almost completely attenuated at large frequencies. This gives a surprising indication that multiple promoter states could filter out the signal oscillation and the noise in the random promoter state selection, as observed in the transcription of a gene activated by p53 in breast carcinoma cells. Our approach may help develop a theoretical framework to integrate coherently the genetic circuit with the promoter states to elucidate the linkage from the activation signal to the temporal profile of transcription outputs.