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 the function is defined.
1. Identify the part of the function that affects the domain:
- The function [tex]\( h(x) \)[/tex] includes a square root, [tex]\( \sqrt{x-7} \)[/tex]. To be able to take the square root of a number, the expression inside the square root must be non-negative. In mathematical terms, it means:
[tex]\[
x - 7 \geq 0
\][/tex]
2. Solve the inequality:
- To find the values of [tex]\( x \)[/tex] that satisfy this inequality, add 7 to both sides:
[tex]\[
x \geq 7
\][/tex]
3. Conclusion:
- 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. This can be written in interval notation as [tex]\([7, \infty)\)[/tex].
Therefore, the correct answer is:
D. [tex]\( x \geq 7 \)[/tex]
1. Identify the part of the function that affects the domain:
- The function [tex]\( h(x) \)[/tex] includes a square root, [tex]\( \sqrt{x-7} \)[/tex]. To be able to take the square root of a number, the expression inside the square root must be non-negative. In mathematical terms, it means:
[tex]\[
x - 7 \geq 0
\][/tex]
2. Solve the inequality:
- To find the values of [tex]\( x \)[/tex] that satisfy this inequality, add 7 to both sides:
[tex]\[
x \geq 7
\][/tex]
3. Conclusion:
- 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. This can be written in interval notation as [tex]\([7, \infty)\)[/tex].
Therefore, the correct answer is:
D. [tex]\( x \geq 7 \)[/tex]
