1. The problem asks to find the domain of the function $$f(x) = \sqrt{-4x - 7}$$. 2. The domain of a square root function requires the expression inside the root to be non-negative because the square root of a negative number is not a real number. 3. Set the inside of the square root greater than or equal to zero: $$-4x - 7 \geq 0$$ 4. Solve the inequality: $$-4x \geq 7$$ 5. Divide both sides by -4, remembering to reverse the inequality sign because we are dividing by a negative number: $$x \leq \cancel{-}\frac{7}{\cancel{-4}}$$ 6. Simplify the fraction: $$x \leq \frac{7}{4}$$ 7. Therefore, the domain of $$f(x)$$ is all real numbers less than or equal to $$\frac{7}{4}$$. 8. In interval notation, the domain is: $$(-\infty, \frac{7}{4}]$$