Assume that we have an activator Ba and a repressor protein Br. We want to obtain an input function such that when a large quantity of Ba is present, the gene is transcriptionally active only if there is no Br, and when low amounts of Ba are present, the gene is transcriptionally inactive (with or without Br).
Write down the reactions among Ba, Br, and the complexes formed with DNA (D) that lead to such an input function. Demonstrate that indeed the set of reactions you picked leads to the desired input function.