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Answer :
To find the domain of the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex], we need to determine the values of [tex]\( x \)[/tex] for which the function is defined.
1. Understand the function:
The function involves a square root. The expression inside the square root [tex]\( \sqrt{x - 7} \)[/tex] must be non-negative because we cannot take the square root of a negative number in the set of real numbers.
2. Set up the inequality:
To ensure that the expression under the square root is non-negative, solve the inequality:
[tex]\[
x - 7 \geq 0
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
Add 7 to both sides of the inequality to isolate [tex]\( x \)[/tex]:
[tex]\[
x \geq 7
\][/tex]
4. Determine the domain:
The solution [tex]\( x \geq 7 \)[/tex] means that the function is defined for all values of [tex]\( x \)[/tex] that are greater than or equal to 7.
Therefore, the domain of the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex] is [tex]\( x \geq 7 \)[/tex]. This corresponds to option D.
1. Understand the function:
The function involves a square root. The expression inside the square root [tex]\( \sqrt{x - 7} \)[/tex] must be non-negative because we cannot take the square root of a negative number in the set of real numbers.
2. Set up the inequality:
To ensure that the expression under the square root is non-negative, solve the inequality:
[tex]\[
x - 7 \geq 0
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
Add 7 to both sides of the inequality to isolate [tex]\( x \)[/tex]:
[tex]\[
x \geq 7
\][/tex]
4. Determine the domain:
The solution [tex]\( x \geq 7 \)[/tex] means that the function is defined for all values of [tex]\( x \)[/tex] that are greater than or equal to 7.
Therefore, the domain of the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex] is [tex]\( x \geq 7 \)[/tex]. This corresponds to option D.
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