To determine the domain of the function [tex]\( y = \sqrt{x + 7} + 5 \)[/tex], we need to ensure that the expression under the square root is non-negative. This is because the square root of a negative number is not defined within the real numbers.
The expression under the square root in our function is [tex]\( x + 7 \)[/tex]. We need this to be greater than or equal to zero:
[tex]\[ x + 7 \geq 0 \][/tex]
Solving for [tex]\( x \)[/tex] gives:
[tex]\[ x \geq -7 \][/tex]
Therefore, the domain of the function [tex]\( y = \sqrt{x + 7} + 5 \)[/tex] is all [tex]\( x \)[/tex] values such that [tex]\( x \geq -7 \)[/tex].
In mathematical notation, the domain is:
[tex]\[ x \geq -7 \][/tex]
Thus, the correct answer is:
[tex]\[ x \geq -7 \][/tex]
