Given circuits for f and f−1, we can create classical reversible circuits Rf, R−1f, Rf−1, R−1f−1 which are shown in the following figure. (Assume that f is a bijection.)In what sequence shall we apply the above circuits in order to implement a reversible circuit that on input x outputs f(x) reversibly? Write your answer in terms of A, B, C, and D using comma as a delimiter. (For example: B,C,A)

To implement a reversible circuit that outputs f(x) on input x, we need to apply the circuits in the following sequence: Rf−1, A, C, B, Rf.

The reason for this sequence is as follows:

1. First, we apply Rf−1. This circuit corresponds to applying the inverse function f−1, which "undoes" the computation of f. This ensures that we start with the original input x.

2. Next, we apply circuit A. This circuit performs the necessary computations to transform the input x into the intermediate result required for f(x).

3. After that, we apply circuit C. This circuit represents the computation of f itself. By applying circuit C, we obtain the desired output f(x).

4. Then, we apply circuit B. This circuit performs the necessary computations to transform the intermediate result of f(x) back into the input format required for the inverse function f−1.

5. Finally, we apply circuit Rf. This circuit corresponds to applying the function f itself but in reverse order, effectively "undoing" the computation of f and bringing us back to the original input x.

Therefore, the correct sequence to implement the reversible circuit is: Rf−1, A, C, B, Rf.