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Answer :
To solve the problem of finding the domain of the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex], follow these steps:
1. Identify the Expression Inside the Square Root:
The square root function [tex]\(\sqrt{x - 7}\)[/tex] is only defined when the expression inside the square root is greater than or equal to zero. This is because you cannot take the square root of a negative number in the set of real numbers.
2. Set Up the Inequality:
To ensure the expression inside the square root is non-negative, set up the inequality:
[tex]\[
x - 7 \geq 0
\][/tex]
3. Solve the Inequality for [tex]\(x\)[/tex]:
Solve the inequality by adding 7 to both sides:
[tex]\[
x \geq 7
\][/tex]
4. Determine the Domain:
The domain of the function is the set of all values of [tex]\(x\)[/tex] that satisfy the inequality above. Thus, [tex]\(x\)[/tex] must be greater than or equal to 7.
Given this reasoning, the domain of the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex] is [tex]\( x \geq 7 \)[/tex].
Therefore, the correct answer is:
- D. [tex]\( x \geq 7 \)[/tex]
1. Identify the Expression Inside the Square Root:
The square root function [tex]\(\sqrt{x - 7}\)[/tex] is only defined when the expression inside the square root is greater than or equal to zero. This is because you cannot take the square root of a negative number in the set of real numbers.
2. Set Up the Inequality:
To ensure the expression inside the square root is non-negative, set up the inequality:
[tex]\[
x - 7 \geq 0
\][/tex]
3. Solve the Inequality for [tex]\(x\)[/tex]:
Solve the inequality by adding 7 to both sides:
[tex]\[
x \geq 7
\][/tex]
4. Determine the Domain:
The domain of the function is the set of all values of [tex]\(x\)[/tex] that satisfy the inequality above. Thus, [tex]\(x\)[/tex] must be greater than or equal to 7.
Given this reasoning, the domain of the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex] is [tex]\( x \geq 7 \)[/tex].
Therefore, the correct answer is:
- D. [tex]\( x \geq 7 \)[/tex]
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