To determine the domain of the function [tex]\( y = 2 \sqrt{x - 6} \)[/tex], we need to consider the conditions under which the function is defined.
1. Square Root Condition: The square root function [tex]\( \sqrt{x-6} \)[/tex] is defined only when the expression inside the square root, [tex]\( x - 6 \)[/tex], is non-negative. This is because the square root of a negative number is not a real number.
2. Non-Negativity: For [tex]\( \sqrt{x-6} \)[/tex] to be valid:
[tex]\[
x - 6 \geq 0
\][/tex]
3. Solving the Inequality: Solving this inequality:
[tex]\[
x - 6 \geq 0 \implies x \geq 6
\][/tex]
Therefore, the domain of the function [tex]\( y = 2 \sqrt{x - 6} \)[/tex] is all [tex]\( x \)[/tex] such that [tex]\( x \geq 6 \)[/tex].
In set notation, the domain can be expressed as:
[tex]\[
[6, \infty)
\][/tex]
So, the correct choice is:
[tex]\( 6 \leq x < \infty \)[/tex].
