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Answer :
To determine the domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex], we need to analyze the expression under the square root. The square root function is only defined for non-negative numbers, which means the expression inside the square root must be greater than or equal to zero.
The expression under the square root is [tex]\( x - 7 \)[/tex]. We set up the inequality:
[tex]\[ x - 7 \geq 0 \][/tex]
Now, we solve this inequality to find the permissible values for [tex]\( x \)[/tex]:
1. Add 7 to both sides of the inequality:
[tex]\[ x - 7 + 7 \geq 0 + 7 \][/tex]
[tex]\[ x \geq 7 \][/tex]
This tells us that [tex]\( x \)[/tex] must be greater than or equal to 7 for [tex]\( h(x) \)[/tex] to be defined. Hence, the domain of the function is all real numbers [tex]\( x \)[/tex] such that [tex]\( x \geq 7 \)[/tex].
Therefore, the correct answer is:
C. [tex]\( x \geq 7 \)[/tex]
The expression under the square root is [tex]\( x - 7 \)[/tex]. We set up the inequality:
[tex]\[ x - 7 \geq 0 \][/tex]
Now, we solve this inequality to find the permissible values for [tex]\( x \)[/tex]:
1. Add 7 to both sides of the inequality:
[tex]\[ x - 7 + 7 \geq 0 + 7 \][/tex]
[tex]\[ x \geq 7 \][/tex]
This tells us that [tex]\( x \)[/tex] must be greater than or equal to 7 for [tex]\( h(x) \)[/tex] to be defined. Hence, the domain of the function is all real numbers [tex]\( x \)[/tex] such that [tex]\( x \geq 7 \)[/tex].
Therefore, the correct answer is:
C. [tex]\( x \geq 7 \)[/tex]
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