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

Sagot :

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].