Unfriendly function to study

Since $f(0) > 0$ and $f(x) \le x + 2 + 3 \pi/2 < 0$ if $x < -2 - 3 \pi/2$, the Intermediate Value Theorem tells you $f$ switches from negative to positive somewhere between $x=-2-3\pi/2$ and $x=0$. The exact point where this occurs can only be determined numerically: it turns out to be approximately $-1.275982661$.