1. The problem asks to find the domain of the function $$f(x) = \sqrt{5 - 2x + bx}$$ with $$b = \frac{1}{2}$$. 2. Substitute $$b = \frac{1}{2}$$ into the function: $$f(x) = \sqrt{5 - 2x + \frac{1}{2}x} = \sqrt{5 - \frac{3}{2}x}$$ 3. The expression inside the square root must be non-negative for the function to be defined: $$5 - \frac{3}{2}x \geq 0$$ 4. Solve the inequality: $$5 \geq \frac{3}{2}x$$ $$x \leq \frac{5}{\frac{3}{2}} = \frac{5 \times 2}{3} = \frac{10}{3} \approx 3.33$$ 5. Therefore, the domain of $$f$$ is all $$x$$ such that: $$x \leq \frac{10}{3}$$ Final answer: $$\boxed{\text{Domain of } f: (-\infty, \frac{10}{3}]}$$