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