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. Identify the restriction: The function involves a square root, [tex]\(\sqrt{x-7}\)[/tex]. Square roots are only defined for non-negative numbers. This means that the expression inside the square root, [tex]\(x-7\)[/tex], must be greater than or equal to 0.
2. Solve the inequality:
[tex]\[
x - 7 \geq 0
\][/tex]
Adding 7 to both sides gives:
[tex]\[
x \geq 7
\][/tex]
3. Determine the domain: The solution to the inequality tells us that the domain of the function is all [tex]\( x \)[/tex] such that [tex]\( x \geq 7 \)[/tex].
Therefore, the correct answer is [tex]\( x \geq 7 \)[/tex], which corresponds to option D.
1. Identify the restriction: The function involves a square root, [tex]\(\sqrt{x-7}\)[/tex]. Square roots are only defined for non-negative numbers. This means that the expression inside the square root, [tex]\(x-7\)[/tex], must be greater than or equal to 0.
2. Solve the inequality:
[tex]\[
x - 7 \geq 0
\][/tex]
Adding 7 to both sides gives:
[tex]\[
x \geq 7
\][/tex]
3. Determine the domain: The solution to the inequality tells us that the domain of the function is all [tex]\( x \)[/tex] such that [tex]\( x \geq 7 \)[/tex].
Therefore, the correct answer is [tex]\( x \geq 7 \)[/tex], which corresponds to option D.
