Answer :
To find the domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex], we need to consider when the square root expression is valid. A square root is only defined for non-negative numbers, meaning the expression inside the square root must be greater than or equal to zero.
So, we have:
[tex]\[
x - 7 \geq 0
\][/tex]
Solving this inequality will help us find the domain:
1. Add 7 to both sides of the inequality:
[tex]\[
x - 7 + 7 \geq 0 + 7
\][/tex]
2. Simplify:
[tex]\[
x \geq 7
\][/tex]
This tells us that for the function [tex]\( h(x) \)[/tex] to have real number outputs, [tex]\( x \)[/tex] must be greater than or equal to 7. Therefore, the domain of the function is all values of [tex]\( x \)[/tex] such that [tex]\( x \geq 7 \)[/tex].
This corresponds to option A: [tex]\( x \geq 7 \)[/tex].
So, we have:
[tex]\[
x - 7 \geq 0
\][/tex]
Solving this inequality will help us find the domain:
1. Add 7 to both sides of the inequality:
[tex]\[
x - 7 + 7 \geq 0 + 7
\][/tex]
2. Simplify:
[tex]\[
x \geq 7
\][/tex]
This tells us that for the function [tex]\( h(x) \)[/tex] to have real number outputs, [tex]\( x \)[/tex] must be greater than or equal to 7. Therefore, the domain of the function is all values of [tex]\( x \)[/tex] such that [tex]\( x \geq 7 \)[/tex].
This corresponds to option A: [tex]\( x \geq 7 \)[/tex].
