To determine the domain of the function [tex]\( y = \sqrt{x + 6} - 7 \)[/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.
Here are the steps to find the domain:
1. Identify the expression inside the square root: The expression inside the square root is [tex]\( x + 6 \)[/tex].
2. Set up the inequality for the expression inside the square root:
Since the square root is defined for non-negative numbers, set up the inequality:
[tex]\[
x + 6 \geq 0
\][/tex]
3. Solve the inequality:
To find the values of [tex]\( x \)[/tex] that satisfy this inequality, solve for [tex]\( x \)[/tex]:
[tex]\[
x + 6 \geq 0 \implies x \geq -6
\][/tex]
4. Conclusion:
The domain of the function is all values of [tex]\( x \)[/tex] such that [tex]\( x \geq -6 \)[/tex]. Therefore, the correct answer is:
[tex]\[
x \geq -6
\][/tex]
This means the function [tex]\( y = \sqrt{x + 6} - 7 \)[/tex] is defined for all [tex]\( x \)[/tex] values that are greater than or equal to [tex]\(-6\)[/tex]. The domain in interval notation is [tex]\([-6, \infty)\)[/tex].
