Answer :
To find the domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex], we need to ensure that the expression inside the square root is non-negative. This is because the square root of a negative number is not defined in the set of real numbers.
Here's a step-by-step explanation:
1. Look at the part inside the square root: [tex]\( x - 7 \)[/tex].
2. Set the expression inside the square root to be greater than or equal to 0, because it must be non-negative:
[tex]\[
x - 7 \geq 0
\][/tex]
3. Solve this inequality for [tex]\( x \)[/tex]:
[tex]\[
x \geq 7
\][/tex]
This result tells us that the domain of the function is all real numbers [tex]\( x \)[/tex] such that [tex]\( x \)[/tex] is greater than or equal to 7.
Therefore, the correct domain for the function [tex]\( h(x) \)[/tex] is option:
- D. [tex]\( x \geq 7 \)[/tex]
Here's a step-by-step explanation:
1. Look at the part inside the square root: [tex]\( x - 7 \)[/tex].
2. Set the expression inside the square root to be greater than or equal to 0, because it must be non-negative:
[tex]\[
x - 7 \geq 0
\][/tex]
3. Solve this inequality for [tex]\( x \)[/tex]:
[tex]\[
x \geq 7
\][/tex]
This result tells us that the domain of the function is all real numbers [tex]\( x \)[/tex] such that [tex]\( x \)[/tex] is greater than or equal to 7.
Therefore, the correct domain for the function [tex]\( h(x) \)[/tex] is option:
- D. [tex]\( x \geq 7 \)[/tex]
