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] for which this function is defined.
1. Look at the square root: The expression inside the square root, [tex]\( \sqrt{x - 7} \)[/tex], must be non-negative because square roots of negative numbers are not defined in the set of real numbers.
2. Set up the inequality: To ensure that the expression [tex]\( x - 7 \)[/tex] is non-negative, we set up the inequality:
[tex]\[
x - 7 \geq 0
\][/tex]
3. Solve the inequality: Solve for [tex]\( x \)[/tex] by adding 7 to both sides of the inequality:
[tex]\[
x \geq 7
\][/tex]
4. Determine the domain: The inequality [tex]\( x \geq 7 \)[/tex] tells us that the values of [tex]\( x \)[/tex] that make the function [tex]\( h(x) \)[/tex] defined are 7 and any number greater than 7.
Thus, the domain of the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex] is all [tex]\( x \)[/tex] such that [tex]\( x \)[/tex] is greater than or equal to 7. Therefore, the correct answer is:
C. [tex]\( x \geq 7 \)[/tex]
1. Look at the square root: The expression inside the square root, [tex]\( \sqrt{x - 7} \)[/tex], must be non-negative because square roots of negative numbers are not defined in the set of real numbers.
2. Set up the inequality: To ensure that the expression [tex]\( x - 7 \)[/tex] is non-negative, we set up the inequality:
[tex]\[
x - 7 \geq 0
\][/tex]
3. Solve the inequality: Solve for [tex]\( x \)[/tex] by adding 7 to both sides of the inequality:
[tex]\[
x \geq 7
\][/tex]
4. Determine the domain: The inequality [tex]\( x \geq 7 \)[/tex] tells us that the values of [tex]\( x \)[/tex] that make the function [tex]\( h(x) \)[/tex] defined are 7 and any number greater than 7.
Thus, the domain of the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex] is all [tex]\( x \)[/tex] such that [tex]\( x \)[/tex] is greater than or equal to 7. Therefore, the correct answer is:
C. [tex]\( x \geq 7 \)[/tex]
