Introduction:
The given problem involves two periodic signals x(t) and y(t) with the same fundamental period of 3 seconds. We are asked to find the fundamental period of the signal z(t) = x(-t) * y(2t + 1).

Explanation:
To determine the fundamental period of the signal z(t), we need to consider the individual periods of the signals x(t) and y(t), and how they affect the overall period of z(t).

Period of x(t):
The signal x(t) has a fundamental period of 3 seconds. This means that x(t + 3) = x(t) for all values of t.

Period of y(t):
Similarly, the signal y(t) also has a fundamental period of 3 seconds. This means that y(t + 3) = y(t) for all values of t.

Period of x(-t):
The signal x(-t) is a time-reversed version of x(t), which means that if we substitute -t for t in x(t), we get x(-t). Since x(t) has a fundamental period of 3 seconds, x(-t) will also have a fundamental period of 3 seconds. This is because x(-t + 3) = x(-(t + 3)) = x(-t) for all values of t.

Period of y(2t + 1):
The signal y(2t + 1) is a time-scaled and time-shifted version of y(t). It is time-scaled by a factor of 2, which means that the period of y(2t + 1) will be half of the period of y(t). Since y(t) has a fundamental period of 3 seconds, y(2t + 1) will have a fundamental period of 3/2 seconds. Additionally, y(2t + 1) is time-shifted by 1 unit to the left. This means that the signal y(2t + 1) will repeat itself 1 unit earlier compared to y(t). However, the overall period of y(2t + 1) remains the same at 3/2 seconds.

Fundamental period of z(t):
The signal z(t) is obtained by multiplying x(-t) and y(2t + 1). Since both x(-t) and y(2t + 1) have the same fundamental period of 3 seconds, the fundamental period of z(t) will also be 3 seconds. This is because z(t + 3) = x(-(t + 3)) * y(2(t + 3) + 1) = x(-t) * y(2t + 1) = z(t) for all values of t.

Conclusion:
The fundamental period of the signal z(t) = x(-t) * y(2t + 1) is 3 seconds. Hence, the correct answer is option D.

X(t) → T = 3 seconds