I've the following function.
$$ G(z) = 2 + \frac{-1+5z^{-1}}{(1-0.5z^{-1})(1-z^{-1})}$$
Calculating it's inverse using $\mathcal Z$-Transform, I get the following function:
$$g[n] = 2\delta[n] + 8u[n] - 9(0.5)^{n}u[n]$$
where $u[n]$ is the unit step and $\delta[n]$ is the impulse.
I tried to do the usual approach to find if a system is time-invariant or not.
That is shifting the input and compute the output $y_1$ and shifting the output $y_2$, compare both; if equal then it's time-invariant, else it isn't time-invariant. However, I don't know what could be the input in this case.
So, I'm supposing that $g[n]$ is the impulse response of the system. But I couldn't find a way to tell if the system is time-invariant or not based on it's impulse response. Maybe I'm missing some key idea here.