To determine the domain of the function [tex]\( y = 2 \sqrt{x-6} \)[/tex], we need to establish the set of all possible values of [tex]\( x \)[/tex] for which the function is defined.
1. The function [tex]\( y = 2 \sqrt{x-6} \)[/tex] involves a square root. For the square root function to be defined, the expression inside the square root must be non-negative.
2. This means we need the expression inside the square root, which is [tex]\( x - 6 \)[/tex], to be greater than or equal to 0.
3. Let's set up the inequality:
[tex]\[
x - 6 \geq 0
\][/tex]
4. Solving for [tex]\( x \)[/tex], add 6 to both sides of the inequality:
[tex]\[
x \geq 6
\][/tex]
5. Thus, the values of [tex]\( x \)[/tex] that make the function [tex]\( y = 2 \sqrt{x-6} \)[/tex] defined are those for which [tex]\( x \)[/tex] is greater than or equal to 6.
Therefore, the domain of the function is:
[tex]\[
6 \leq x < \infty
\][/tex]
The correct answer is:
[tex]\[ 6 \leq x < \infty \][/tex]
