Answer :
To determine the domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex], we need to identify the values of [tex]\( x \)[/tex] for which the function is defined. The critical part here is the square root function [tex]\( \sqrt{x-7} \)[/tex], which requires the expression inside the square root to be non-negative. This means:
[tex]\[
x - 7 \geq 0
\][/tex]
Solving this inequality involves adding 7 to both sides:
[tex]\[
x \geq 7
\][/tex]
This means the function is defined for all [tex]\( x \)[/tex] that are greater than or equal to 7. Therefore, the domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex] is:
[tex]\[
x \geq 7
\][/tex]
Looking at the given options:
- A. [tex]\( x \geq 7 \)[/tex]
- B. [tex]\( x \leq 5 \)[/tex]
- C. [tex]\( x \leq -7 \)[/tex]
- D. [tex]\( x \geq 5 \)[/tex]
The correct choice that matches the domain we found is Option A: [tex]\( x \geq 7 \)[/tex].
[tex]\[
x - 7 \geq 0
\][/tex]
Solving this inequality involves adding 7 to both sides:
[tex]\[
x \geq 7
\][/tex]
This means the function is defined for all [tex]\( x \)[/tex] that are greater than or equal to 7. Therefore, the domain of the function [tex]\( h(x) = \sqrt{x-7} + 5 \)[/tex] is:
[tex]\[
x \geq 7
\][/tex]
Looking at the given options:
- A. [tex]\( x \geq 7 \)[/tex]
- B. [tex]\( x \leq 5 \)[/tex]
- C. [tex]\( x \leq -7 \)[/tex]
- D. [tex]\( x \geq 5 \)[/tex]
The correct choice that matches the domain we found is Option A: [tex]\( x \geq 7 \)[/tex].
