1. We need the domain of $f(x)=\frac{1}{\sqrt{|x|-x}}.$

2. Since the expression is in the denominator under a square root, we need:

   • the quantity inside the square root to be defined, and
   • the denominator must be nonzero.

   So we require $|x|-x>0.$

3. Now evaluate piecewise using the definition of modulus.

   Case 1: $x\ge 0$ $|x|=x$ so $|x|-x=x-x=0.$ Then $\sqrt{|x|-x}=\sqrt{0}=0,$ and the denominator becomes zero, which is not allowed.

   Hence, no $x\ge 0$ belongs to the domain.

4. Case 2: $x<0$ $|x|=-x$ so $|x|-x=-x-x=-2x.$ Since $x<0$, we have $-2x>0$. Therefore the square root is real and nonzero.

   Hence all $x<0$ are allowed.

5. Therefore, the domain is $(-\infty,0).$

6. Comparing with the options:

   • A: $(0,\infty)$ ❌
   • B: $(-\infty,0)$ ✅
   • C: $(-\infty,\infty)-\{0\}$ ❌
   • D: $(-\infty,\infty)$ ❌

So the correct option is B.