What is the domain of the function [tex]y=2 \sqrt{x-6}[/tex]?

A. [tex]-\infty \ \textless \ x \ \textless \ \infty[/tex]
B. [tex]0 \leq x \ \textless \ \infty[/tex]
C. [tex]3 \leq x \ \textless \ \infty[/tex]
D. [tex]6 \leq x \ \textless \ \infty[/tex]



Answer :

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]