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Answer :
To find the domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex], we need to consider when the expression under the square root is defined. Specifically, the square root function is only defined for non-negative numbers.
Let's break it down step-by-step:
1. Identify the Expression Under the Square Root:
The expression under the square root is [tex]\( x - 7 \)[/tex].
2. Set Up an Inequality for the Square Root's Domain:
For the square root to be defined, we need:
[tex]\[
x - 7 \geq 0
\][/tex]
3. Solve the Inequality:
To solve this inequality, add 7 to both sides:
[tex]\[
x \geq 7
\][/tex]
4. Conclusion on the Domain:
The inequality [tex]\( x \geq 7 \)[/tex] represents the set of all [tex]\( x \)[/tex]-values for which the function [tex]\( h(x) \)[/tex] will produce real numbers. Therefore, the domain of [tex]\( h \)[/tex] is all [tex]\( x \)[/tex] such that [tex]\( x \geq 7 \)[/tex].
Based on this step-by-step analysis, the correct answer is option B: [tex]\( x \geq 7 \)[/tex].
Let's break it down step-by-step:
1. Identify the Expression Under the Square Root:
The expression under the square root is [tex]\( x - 7 \)[/tex].
2. Set Up an Inequality for the Square Root's Domain:
For the square root to be defined, we need:
[tex]\[
x - 7 \geq 0
\][/tex]
3. Solve the Inequality:
To solve this inequality, add 7 to both sides:
[tex]\[
x \geq 7
\][/tex]
4. Conclusion on the Domain:
The inequality [tex]\( x \geq 7 \)[/tex] represents the set of all [tex]\( x \)[/tex]-values for which the function [tex]\( h(x) \)[/tex] will produce real numbers. Therefore, the domain of [tex]\( h \)[/tex] is all [tex]\( x \)[/tex] such that [tex]\( x \geq 7 \)[/tex].
Based on this step-by-step analysis, the correct answer is option B: [tex]\( x \geq 7 \)[/tex].
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