Westonci.ca is the ultimate Q&A platform, offering detailed and reliable answers from a knowledgeable community. Get detailed and precise answers to your questions from a dedicated community of experts on our Q&A platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
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].
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].
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.