Question

Find DRG of
$f(x) = \frac{1 - x + 2\sqrt{-x}}{1 + x}$

Answer 1

$x \le 0,\; x \neq -1$

Answer 2

Step 1. For the expression $\sqrt{-x}$ to be real, we require

$$-x \ge 0 \quad\Longrightarrow\quad x \le 0.$$

Step 2. The denominator $1 + x$ must be nonzero:

$$1 + x \neq 0 \quad\Longrightarrow\quad x \neq -1.$$

Conclusion:
Combining both conditions gives the domain

$$\boxed{x \le 0,\; x \neq -1},$$

i.e.\ the union $(-\infty,-1)\,\cup\,(-1,0].$