To determine the domain of the function [tex]\( y = 2 \sqrt{x-6} \)[/tex], we need to ensure that all operations within the function are valid for real numbers.
1. The relevant part of the function is the square root [tex]\( \sqrt{x-6} \)[/tex]. For the square root function to be defined, the expression inside the square root must be non-negative because the square root of a negative number is not a real number.
2. Set up the inequality to ensure the argument of the square root is non-negative:
[tex]\[
x - 6 \geq 0
\][/tex]
3. Solve this inequality for [tex]\( x \)[/tex]:
[tex]\[
x \geq 6
\][/tex]
4. This implies that [tex]\( x \)[/tex] must be at least 6. Therefore, the domain is all [tex]\( x \)[/tex] such that [tex]\( x \ge 6 \)[/tex].
5. Expressing this in interval notation, the domain is:
[tex]\[
[6, \infty)
\][/tex]
Looking at the answer choices provided:
1. [tex]\( -\infty < x < \infty \)[/tex]
2. [tex]\( 0 \leq x < \infty \)[/tex]
3. [tex]\( 3 \leq x < \infty \)[/tex]
4. [tex]\( 6 \leq x < \infty \)[/tex]
The correct answer is:
[tex]\[
6 \leq x < \infty
\][/tex]
So, the domain of the function [tex]\( y = 2 \sqrt{x-6} \)[/tex] is given by the fourth option: [tex]\( 6 \leq x < \infty \)[/tex].
