To determine the domain of the function [tex]\( y = \sqrt{x + 7} + 5 \)[/tex], we need to ensure that the expression inside the square root is non-negative. This is because the square root function is only defined for non-negative numbers (i.e., it cannot produce real numbers for negative inputs).
Given the function [tex]\( y = \sqrt{x + 7} + 5 \)[/tex], we focus on the expression inside the square root, which is [tex]\( x + 7 \)[/tex].
To ensure [tex]\( \sqrt{x + 7} \)[/tex] is defined, the following inequality must hold:
[tex]\[ x + 7 \geq 0 \][/tex]
Now, solve this inequality for [tex]\( x \)[/tex]:
[tex]\[
x + 7 \geq 0
\][/tex]
[tex]\[
x \geq -7
\][/tex]
Therefore, the values of [tex]\( x \)[/tex] that make the expression under the square root non-negative are all [tex]\( x \)[/tex] such that [tex]\( x \geq -7 \)[/tex].
Hence, the domain of the function [tex]\( y = \sqrt{x + 7} + 5 \)[/tex] is:
[tex]\[ \boxed{x \geq -7} \][/tex]
