1. **State the problem:** We are given the function $f(x) = \frac{1}{2} \sqrt{-4x}$ and need to understand its domain and simplify it if possible. 2. **Recall the domain rule for square roots:** The expression inside the square root must be greater than or equal to zero for the function to be real-valued. So, we require: $$-4x \geq 0$$ 3. **Solve the inequality:** $$-4x \geq 0$$ Divide both sides by -4, remembering to reverse the inequality sign because we divide by a negative number: $$\cancel{-4}x \leq \cancel{-4}0$$ which simplifies to: $$x \leq 0$$ 4. **Simplify the function:** Inside the square root, factor out the negative sign: $$\sqrt{-4x} = \sqrt{4(-x)} = \sqrt{4} \sqrt{-x} = 2 \sqrt{-x}$$ 5. **Substitute back into $f(x)$:** $$f(x) = \frac{1}{2} \times 2 \sqrt{-x} = \sqrt{-x}$$ 6. **Final function and domain:** $$f(x) = \sqrt{-x} \quad \text{with domain} \quad x \leq 0$$ This means $f(x)$ is defined for all $x$ less than or equal to zero, and the function simplifies to $\sqrt{-x}$.