Answer :
To determine the domain of the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex], we need to consider when 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 are the steps to find the domain:
1. Identify the expression inside the square root: For the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex], the expression inside the square root is [tex]\( x - 7 \)[/tex].
2. Set the expression greater than or equal to zero: We need [tex]\( x - 7 \geq 0 \)[/tex]. This ensures that the expression inside the square root is non-negative, which is necessary for the function to be defined.
3. Solve the inequality:
- [tex]\( x - 7 \geq 0 \)[/tex]
- Add 7 to both sides: [tex]\( x \geq 7 \)[/tex]
4. State the domain based on the inequality: The solution [tex]\( x \geq 7 \)[/tex] describes the domain of the function. This means all [tex]\( x \)[/tex]-values greater than or equal to 7 will make the function [tex]\( h(x) \)[/tex] defined.
Thus, the domain of the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex] is [tex]\( x \geq 7 \)[/tex]. The correct answer, corresponding to this domain, is:
D. [tex]\( x \geq 7 \)[/tex]
Here are the steps to find the domain:
1. Identify the expression inside the square root: For the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex], the expression inside the square root is [tex]\( x - 7 \)[/tex].
2. Set the expression greater than or equal to zero: We need [tex]\( x - 7 \geq 0 \)[/tex]. This ensures that the expression inside the square root is non-negative, which is necessary for the function to be defined.
3. Solve the inequality:
- [tex]\( x - 7 \geq 0 \)[/tex]
- Add 7 to both sides: [tex]\( x \geq 7 \)[/tex]
4. State the domain based on the inequality: The solution [tex]\( x \geq 7 \)[/tex] describes the domain of the function. This means all [tex]\( x \)[/tex]-values greater than or equal to 7 will make the function [tex]\( h(x) \)[/tex] defined.
Thus, the domain of the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex] is [tex]\( x \geq 7 \)[/tex]. The correct answer, corresponding to this domain, is:
D. [tex]\( x \geq 7 \)[/tex]
