We have the relationship between input and output: $Y(k)=|x(k+1)| + x(k)+ kx(k-5)$.
Find the output of the system when $x(k)=d(k)$.
What does this output represent?
Show if the system is linear, invariant, causal, stable, etc
Edit : I am familiar with sistem properties ,but what I mostly dont understand in the question is how to find the output when when $x(k)=d(k)$ and what it represents.To be more clear,what does it mean when when $x(k)=d(k)$.
If $d(k)=\delta(k)$, i.e. the discrete unit impulse, then the response of the system to the input $x(k)=\delta(k)$ is its impulse response. However, since the system is not linear and time-invariant, it is not fully characterized by its impulse response. These are the system's properties:
it is non-linear, because the absolute value is used to compute the output signal
it is time-varying, because the time index $k$ appears as a multiplicative constant in the system equation.
it is non-causal, because in order to compute $y(k)$ the input at time $k+1$ must be known.
it is not BIBO-stable, because even for a bounded input, the output is not bounded due to the multiplicative constant $k$ which grows without bounds.