To determine the domain of the function [tex]\( y = -4 \sqrt{-3x - 18} + 7 \)[/tex], we need to ensure that the expression inside the square root is non-negative. This is because the square root function is defined only for non-negative arguments.
Given the function [tex]\( y = -4 \sqrt{-3x - 18} + 7 \)[/tex], consider the expression inside the square root:
[tex]\[ -3x - 18 \][/tex]
We need this expression to be greater than or equal to zero for the square root to be defined:
[tex]\[ -3x - 18 \geq 0 \][/tex]
Solving this inequality step-by-step gives:
1. Add 18 to both sides of the inequality:
[tex]\[ -3x \geq 18 \][/tex]
2. Divide both sides by -3. Remember that dividing or multiplying both sides of an inequality by a negative number reverses the inequality:
[tex]\[ x \leq -6 \][/tex]
Hence, the domain of the function [tex]\( y = -4 \sqrt{-3x - 18} + 7 \)[/tex] is all [tex]\( x \)[/tex] such that [tex]\( x \leq -6 \)[/tex].
Thus, the correct answer is:
d) [tex]\( x \leq -6 \)[/tex]
