Answer :
To find the domain of the function [tex]\( h(x) = \sqrt{x - 7} + 5 \)[/tex], we need to focus on the expression inside the square root, [tex]\( \sqrt{x - 7} \)[/tex].
1. Understand the square root requirement: The square root function [tex]\(\sqrt{y}\)[/tex] is defined only when the value inside the square root (in this case, [tex]\( y = x - 7 \)[/tex]) is non-negative. This means that [tex]\( x - 7 \)[/tex] must be greater than or equal to zero:
[tex]\[
x - 7 \geq 0
\][/tex]
2. Solve the inequality: To find the values of [tex]\( x \)[/tex] that satisfy the inequality, solve for [tex]\( x \)[/tex]:
[tex]\[
x \geq 7
\][/tex]
3. Conclusion about the domain: The domain of the function [tex]\( h(x) \)[/tex] is all [tex]\( x \)[/tex] such that [tex]\( x \)[/tex] is greater than or equal to 7. This ensures that the expression inside the square root is non-negative and thus valid.
Given the options:
- A. [tex]\( x \leq 5 \)[/tex]
- B. [tex]\( x \geq 5 \)[/tex]
- C. [tex]\( x \leq -7 \)[/tex]
- D. [tex]\( x \geq 7 \)[/tex]
The correct answer, based on the above analysis, is D. [tex]\( x \geq 7 \)[/tex].
1. Understand the square root requirement: The square root function [tex]\(\sqrt{y}\)[/tex] is defined only when the value inside the square root (in this case, [tex]\( y = x - 7 \)[/tex]) is non-negative. This means that [tex]\( x - 7 \)[/tex] must be greater than or equal to zero:
[tex]\[
x - 7 \geq 0
\][/tex]
2. Solve the inequality: To find the values of [tex]\( x \)[/tex] that satisfy the inequality, solve for [tex]\( x \)[/tex]:
[tex]\[
x \geq 7
\][/tex]
3. Conclusion about the domain: The domain of the function [tex]\( h(x) \)[/tex] is all [tex]\( x \)[/tex] such that [tex]\( x \)[/tex] is greater than or equal to 7. This ensures that the expression inside the square root is non-negative and thus valid.
Given the options:
- A. [tex]\( x \leq 5 \)[/tex]
- B. [tex]\( x \geq 5 \)[/tex]
- C. [tex]\( x \leq -7 \)[/tex]
- D. [tex]\( x \geq 7 \)[/tex]
The correct answer, based on the above analysis, is D. [tex]\( x \geq 7 \)[/tex].
