Answer :
To find the domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex], we need to determine the values of [tex]\( x \)[/tex] that make the expression inside the square root, [tex]\( x-7 \)[/tex], non-negative (greater than or equal to zero). This is because the square root of a negative number is not defined in the set of real numbers.
Let's follow these steps:
1. Set up the inequality:
The expression inside the square root is [tex]\( x - 7 \)[/tex]. We need this to be non-negative:
[tex]\[
x - 7 \geq 0
\][/tex]
2. Solve the inequality for [tex]\( x \)[/tex]:
To solve for [tex]\( x \)[/tex], add 7 to both sides of the inequality:
[tex]\[
x \geq 7
\][/tex]
3. Conclusion about the domain:
The inequality [tex]\( x \geq 7 \)[/tex] means that [tex]\( x \)[/tex] can be 7 or any number greater than 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].
Thus, the correct answer is A. [tex]\( x \geq 7 \)[/tex].
Let's follow these steps:
1. Set up the inequality:
The expression inside the square root is [tex]\( x - 7 \)[/tex]. We need this to be non-negative:
[tex]\[
x - 7 \geq 0
\][/tex]
2. Solve the inequality for [tex]\( x \)[/tex]:
To solve for [tex]\( x \)[/tex], add 7 to both sides of the inequality:
[tex]\[
x \geq 7
\][/tex]
3. Conclusion about the domain:
The inequality [tex]\( x \geq 7 \)[/tex] means that [tex]\( x \)[/tex] can be 7 or any number greater than 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].
Thus, the correct answer is A. [tex]\( x \geq 7 \)[/tex].
