$h=2\pi \cdot \chi_{(-4,4)} -3\pi \cdot \chi_{[-\sqrt{2},\sqrt{2})} + 4\pi + \chi_{(-1,1)}$
I am unsure how to rewrite this using disjoint intervals. I know that $(-1,1)\in [-4,4)$ and $(-1,1)\in [-\sqrt{2},\sqrt{2})$. Finding the new coefficients is what is causing my problems. Is there an easy way to do this?
My attempt at disjoint intervals:
$h=$ __ $\pi \cdot \chi_{(-4,-\sqrt{2}]} +$ _ $\pi \cdot \chi_{(-\sqrt{2}, -1)} +$ _ $\pi \cdot \chi_{(-1,1)} +$ ____ $\pi \cdot \chi_{(1,\sqrt{2})} +$ ___ $\pi \cdot \chi_{(\sqrt{2}, 4)}$
Method #1: If you have a two intervals you can always write them as the union at most three disjoint intervals. You can repeat the process to write any finite collection of intervals as a disjoint collection. Now evaluate $h$ on these disjoint intervals to get the required form.
Method #2: Determine the range of $h$, in this case the range has four elements $r_1,...,r_4$. Then determine $h^{-1}(\{r_k\})$ which will be the union of disjoint intervals.
The disjoint intervals in this question are $I_0 = (-\infty,4]$, $I_1= (-4,-\sqrt{2})$, $I_2=[-\sqrt{2}, -1]$, $I_3 = (-1,1)$, $I_4=[1,2)$, $I_5 = [2,4)$ and $I_6 = [4,\infty)$.
It is straightforward to evaluate $h$ at any point of each of these intervals.