Answer :
To determine the domain of the function \( y = \sqrt{x + 6} - 7 \), let's follow these steps:
1. Identify the expression inside the square root: The function involves a square root, which is \( \sqrt{x + 6} \).
2. Determine the condition for the square root: The square root function is defined only when the radicand (the expression inside the square root) is non-negative. In other words, the expression inside the square root must be greater than or equal to zero.
[tex]\[ x + 6 \geq 0 \][/tex]
3. Solve the inequality for \( x \):
[tex]\[ x + 6 \geq 0 \][/tex]
Subtract 6 from both sides:
[tex]\[ x \geq -6 \][/tex]
Based on this solution, the domain of the function \( y = \sqrt{x + 6} - 7 \) is all values of \( x \) such that \( x \geq -6 \).
Thus, the correct answer is:
[tex]\[ x \geq -6 \][/tex]
1. Identify the expression inside the square root: The function involves a square root, which is \( \sqrt{x + 6} \).
2. Determine the condition for the square root: The square root function is defined only when the radicand (the expression inside the square root) is non-negative. In other words, the expression inside the square root must be greater than or equal to zero.
[tex]\[ x + 6 \geq 0 \][/tex]
3. Solve the inequality for \( x \):
[tex]\[ x + 6 \geq 0 \][/tex]
Subtract 6 from both sides:
[tex]\[ x \geq -6 \][/tex]
Based on this solution, the domain of the function \( y = \sqrt{x + 6} - 7 \) is all values of \( x \) such that \( x \geq -6 \).
Thus, the correct answer is:
[tex]\[ x \geq -6 \][/tex]
