Final answer:
The function of a trans-acting transcription factor binding to cis-acting sites in a prokaryotic genome can be that of an activator or a repressor, depending on the operon, specific DNA sequence, and environmental conditions.
Step-by-step explanation:
When a trans-acting transcription factor binds to cis-acting sites in a prokaryotic genome, its function at these sites can either be as an activator or a repressor of transcription. Activators increase gene transcription by facilitating RNA polymerase binding to the promoter; repressors suppress transcription by preventing RNA polymerase from transcribing the operon.
The specific sequence of nucleotides at the binding site (often at an operator or promoter) and the presence of additional molecules such as ligands or inducers determine the mode of regulation. For instance, in the case of the lac operon, the absence of glucose and the presence of lactose results in transcription activation, whereas in the trp operon, the presence of tryptophan leads to the repression of transcription.