Question

what is the domain of the function $y = \sqrt{x + 6} - 7$?
\\(\circ\\) $x \geq -7$
\\(\circ\\) $x \geq -6$
\\(\circ\\) $x \geq 6$
\\(\circ\\) $x \geq 7$

Explanation:

Step1: Recall the domain rule for square roots

For a square root function \(\sqrt{u}\), the expression inside the square root (the radicand) must be non - negative, i.e., \(u\geq0\). In the function \(y = \sqrt{x + 6}-7\), the radicand is \(x + 6\).

Step2: Set up the inequality for the radicand

We set up the inequality \(x+6\geq0\).

Step3: Solve the inequality

To solve \(x + 6\geq0\), we subtract 6 from both sides of the inequality. Using the subtraction property of inequalities (if \(a\geq b\), then \(a - c\geq b - c\) for any real number \(c\)), we get \(x\geq - 6\).

Answer:

\(x\geq - 6\) (corresponding to the option with text \(x\geq - 6\))