To determine the domain of the function [tex]\( y = \sqrt{x + 7} + 5 \)[/tex], we need to identify the set of all possible values for [tex]\( x \)[/tex] that make the function well-defined.
The function involves a square root, [tex]\(\sqrt{x + 7}\)[/tex]. For the square root function to be defined, its radicand (the expression inside the square root) must be non-negative. Therefore, we need:
[tex]\[
x + 7 \geq 0
\][/tex]
Solving this inequality for [tex]\( x \)[/tex], we subtract 7 from both sides:
[tex]\[
x \geq -7
\][/tex]
This means that [tex]\( x \)[/tex] must be greater than or equal to [tex]\(-7\)[/tex] for the expression [tex]\(\sqrt{x + 7}\)[/tex] to be defined. Once we have this condition, the entire function [tex]\( y = \sqrt{x + 7} + 5 \)[/tex] will be defined as long as [tex]\( x \geq -7 \)[/tex].
Thus, the domain of the function [tex]\( y = \sqrt{x + 7} + 5 \)[/tex] is:
[tex]\[
x \geq -7
\][/tex]
Therefore, the correct answer is:
[tex]\[
x \geq -7
\][/tex]
