To determine the domain of the function [tex]\( y = 2 \sqrt{x - 6} \)[/tex], we need to consider where the function [tex]\( y \)[/tex] is defined and produces real numbers.
A square root function, [tex]\( \sqrt{u} \)[/tex], is defined only when the expression under the square root, [tex]\( u \)[/tex], is non-negative (i.e., [tex]\( u \geq 0 \)[/tex]) because the square root of a negative number is not a real number.
In the given function [tex]\( y = 2 \sqrt{x - 6} \)[/tex]:
1. Identify the expression under the square root: [tex]\( x - 6 \)[/tex].
2. Set up an inequality to ensure the expression is non-negative:
[tex]\[
x - 6 \geq 0
\][/tex]
3. Solve the inequality for [tex]\( x \)[/tex]:
[tex]\[
x \geq 6
\][/tex]
Therefore, the domain of the function is all [tex]\( x \)[/tex] values that are greater than or equal to 6. In interval notation, this is written as [tex]\( [6, \infty) \)[/tex].
In conclusion, the domain of the function [tex]\( y = 2 \sqrt{x - 6} \)[/tex] is:
[tex]\[ \boxed{6 \leq x < \infty} \][/tex]
