Explore Westonci.ca, the leading Q&A site where experts provide accurate and helpful answers to all your questions. Discover detailed solutions to your questions from a wide network of experts on our comprehensive Q&A platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.

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].
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.