Interpretation of system response to unit impulse
In the figure that you have posted, the right plot shows the impulse response $h[n]$ according to the convention that only its nonzero samples are displayed. And therefore, it's understood that all the remaning samples of $h[n]$ are zero unless otherwise stated explicitly.

This means that $h[0]=0$, $h[1]=0$,...,$h[12]=0$. Which can be stated in the following:

$$ h[n] = \begin{cases}
0 &amp;,&amp; \text{ for } n \le 4 \\
\alpha_n &amp;,&amp; \text{ for } 5 \le n \le 11 \\
0 &amp;,&amp; \text{ for } n \ge 12 \\
\end{cases}
$$

In this setting the domain of support for the finite length sequence $h[n]$ is stated as $n \in [5,11]$ , which means that $h[n]$ is certainly zero for $n$ out of this interval, while it can have nonzero as well as zero values inside this interval.