Answer :
To determine 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 how to solve it step by step:
1. Identify the expression inside the square root:
The expression inside the square root is [tex]\( x - 7 \)[/tex].
2. Set the expression inside the square root to be non-negative:
We need [tex]\( x - 7 \geq 0 \)[/tex].
3. Solve the inequality for [tex]\( x \)[/tex]:
[tex]\[
x - 7 \geq 0
\][/tex]
Adding 7 to both sides gives:
[tex]\[
x \geq 7
\][/tex]
4. Conclusion:
The domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex] is [tex]\( x \geq 7 \)[/tex].
Thus, the correct answer is D. [tex]\( x \geq 7 \)[/tex].
Here's how to solve it step by step:
1. Identify the expression inside the square root:
The expression inside the square root is [tex]\( x - 7 \)[/tex].
2. Set the expression inside the square root to be non-negative:
We need [tex]\( x - 7 \geq 0 \)[/tex].
3. Solve the inequality for [tex]\( x \)[/tex]:
[tex]\[
x - 7 \geq 0
\][/tex]
Adding 7 to both sides gives:
[tex]\[
x \geq 7
\][/tex]
4. Conclusion:
The domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex] is [tex]\( x \geq 7 \)[/tex].
Thus, the correct answer is D. [tex]\( x \geq 7 \)[/tex].
