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Answer :
To determine the domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex], we need to ensure that the expression inside the square root, [tex]\( x-7 \)[/tex], is non-negative. This is because the square root of a negative number is not defined in the set of real numbers.
Here's how we find the domain:
1. Set Up the Inequality:
We want the expression inside the square root to be greater than or equal to zero:
[tex]\[
x - 7 \geq 0
\][/tex]
2. Solve the Inequality:
To solve for [tex]\( x \)[/tex], add 7 to both sides of the inequality:
[tex]\[
x \geq 7
\][/tex]
3. Determine the Domain:
The solution to the inequality [tex]\( x \geq 7 \)[/tex] means that [tex]\( x \)[/tex] can be any number greater than or equal to 7. This ensures the expression inside the square root is non-negative.
Therefore, the domain of the function is all real numbers [tex]\( x \)[/tex] such that [tex]\( x \geq 7 \)[/tex].
So, the correct answer is:
- [tex]\(\boxed{D. \, x \geq 7}\)[/tex]
Here's how we find the domain:
1. Set Up the Inequality:
We want the expression inside the square root to be greater than or equal to zero:
[tex]\[
x - 7 \geq 0
\][/tex]
2. Solve the Inequality:
To solve for [tex]\( x \)[/tex], add 7 to both sides of the inequality:
[tex]\[
x \geq 7
\][/tex]
3. Determine the Domain:
The solution to the inequality [tex]\( x \geq 7 \)[/tex] means that [tex]\( x \)[/tex] can be any number greater than or equal to 7. This ensures the expression inside the square root is non-negative.
Therefore, the domain of the function is all real numbers [tex]\( x \)[/tex] such that [tex]\( x \geq 7 \)[/tex].
So, the correct answer is:
- [tex]\(\boxed{D. \, x \geq 7}\)[/tex]
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