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Answer :
To determine the domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex], we need to focus on the expression inside the square root, [tex]\(\sqrt{x-7}\)[/tex].
Here's a step-by-step guide to finding the domain:
1. Understand the Square Root Requirement:
- For a square root function to be defined, the expression inside the square root must be non-negative (i.e., it cannot be negative).
- Therefore, we need the expression [tex]\( x - 7 \)[/tex] to be greater than or equal to zero.
2. Set Up the Inequality:
- We form the inequality: [tex]\( x - 7 \geq 0 \)[/tex].
3. Solve the Inequality:
- To solve for [tex]\( x \)[/tex], we simply add 7 to both sides of the inequality:
[tex]\[
x \geq 7
\][/tex]
4. Determine the Domain:
- The inequality [tex]\( x \geq 7 \)[/tex] tells us the values of [tex]\( x \)[/tex] for which the function [tex]\( h(x) \)[/tex] is defined.
- This means the function can take any value of [tex]\( x \)[/tex] as long as [tex]\( x \)[/tex] is 7 or greater.
5. Conclude the Solution:
- Therefore, the domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex] is all real numbers [tex]\( x \)[/tex] such that [tex]\( x \geq 7 \)[/tex].
Accordingly, the correct answer is:
B. [tex]\( x \geq 7 \)[/tex]
Here's a step-by-step guide to finding the domain:
1. Understand the Square Root Requirement:
- For a square root function to be defined, the expression inside the square root must be non-negative (i.e., it cannot be negative).
- Therefore, we need the expression [tex]\( x - 7 \)[/tex] to be greater than or equal to zero.
2. Set Up the Inequality:
- We form the inequality: [tex]\( x - 7 \geq 0 \)[/tex].
3. Solve the Inequality:
- To solve for [tex]\( x \)[/tex], we simply add 7 to both sides of the inequality:
[tex]\[
x \geq 7
\][/tex]
4. Determine the Domain:
- The inequality [tex]\( x \geq 7 \)[/tex] tells us the values of [tex]\( x \)[/tex] for which the function [tex]\( h(x) \)[/tex] is defined.
- This means the function can take any value of [tex]\( x \)[/tex] as long as [tex]\( x \)[/tex] is 7 or greater.
5. Conclude the Solution:
- Therefore, the domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex] is all real numbers [tex]\( x \)[/tex] such that [tex]\( x \geq 7 \)[/tex].
Accordingly, the correct answer is:
B. [tex]\( x \geq 7 \)[/tex]
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