Inverse System:

The given system is y(t) = 2x(t). We need to find the inverse system of this given system.

The inverse system is obtained by swapping the input and output of the given system. Therefore, the output of the inverse system becomes the input of the given system, and vice versa.

To find the inverse system, we can express the given system as y(t) = H[x(t)], where H represents the system or operator.

Now, let's swap the input and output variables:

x(t) = H^(-1)[y(t)]

This equation represents the inverse system, where x(t) is the output of the inverse system and y(t) is the input.

Comparing this equation with the given options, we can see that option A satisfies the equation. Therefore, the inverse system is given by y(t) = 0.5x(t).

Explanation of Option A:

Option A states that the output of the inverse system, y(t), is equal to 0.5 times the input of the inverse system, x(t).

This means that for a given input x(t), the output of the inverse system will be half of the input. In other words, the inverse system attenuates the input signal by a factor of 0.5.

If we apply this inverse system to the output of the given system y(t) = 2x(t), we should obtain the original input signal x(t) as the output.

Let's verify this:

Given system: y(t) = 2x(t)

Inverse system (Option A): y(t) = 0.5x(t)

If we apply the given system followed by the inverse system, we get:

y(t) = 2x(t) (Given system)
y(t) = 0.5(2x(t)) (Inverse system)
y(t) = x(t)

As we can see, the output of the inverse system is equal to the original input signal x(t). Hence, option A represents the correct inverse system.

Final Answer: Option A