To determine the domain of the function [tex]\( y = \sqrt{x + 7} + 5 \)[/tex], we need to identify the range of [tex]\( x \)[/tex] values for which the function is defined and produces real numbers.
1. The function includes a square root, [tex]\(\sqrt{x + 7}\)[/tex]. The expression inside the square root must be non-negative because the square root of a negative number is not a real number.
2. Therefore, the inequality [tex]\( x + 7 \geq 0 \)[/tex] must be satisfied.
3. Solving this inequality for [tex]\( x \)[/tex]:
[tex]\[ x + 7 \geq 0 \][/tex]
[tex]\[ x \geq -7 \][/tex]
4. This indicates that the domain of [tex]\( y = \sqrt{x + 7} + 5 \)[/tex] is all [tex]\( x \)[/tex] values that are greater than or equal to [tex]\(-7\)[/tex].
So, the correct answer is:
[tex]\[ x \geq -7 \][/tex]
In interval notation, the domain is [tex]\([-7, \infty)\)[/tex]. Hence, the appropriate choice from the given options is:
[tex]\[ x \geq -7 \][/tex]
