Final answer:
The continuous-time system given by y(t) = x(t⁻²) + x(2-t) is not memoryless, not time invariant, appears to be linear, is not causal, and we cannot definitively determine its stability without additional information on x(t).
Step-by-step explanation:
The question revolves around determining the properties of a specific continuous-time system given by the equation y(t) = x(t⁻²) + x(2-t). Let's analyze this system with respect to the properties introduced in the chapter:
Memoryless: A system is memoryless if its output for any given input at time t is dependent only on the input at that time and not on any past inputs. This system is not memoryless, because the output y(t) depends on x(t⁻²) and x(2-t), which are input values at times other than the current time t.
Time Invariant: A system is time invariant if a time shift in the input signal results in an identical time shift in the output signal. The given system is not time invariant, because shifting the input x(t) in time would not yield a simple time shift in y(t) due to the squared and subtracted time variable present in the arguments of x(t).
Linear: A system is linear if it satisfies the principles of superposition and homogeneity. The given system appears to be linear, as the output is the sum of two transformed versions of the input, and there are no products or powers of x(t) that would violate linearity.
Causal: A system is causal if its output at any time depends only on inputs from the current or past time, not future inputs. The given system is not causal because it involves x(2-t), which for t > 2 would require knowledge of future inputs.
Stable: A system is stable if bounded input leads to bounded output. Without more information about x(t), we cannot definitively determine stability for this system, as the output heavily depends on the behavior of x(t).