Question

which function has the domain \\(x \ge -11\\)?

\\(y = \sqrt{x + 11} + 5\\)
\\(y = \sqrt{x - 11} + 5\\)
\\(y = \sqrt{x + 5} - 11\\)
\\(y = \sqrt{x + 5} + 11\\)

Explanation:

Set up the domain inequality for a square root function

For a real-valued square root function \(y = \sqrt{g(x)} + c\), the radicand must be non-negative:

$$g(x) \ge 0$$

Analyze each option's domain

• For \(y = \sqrt{x + 11} + 5\):

$$x + 11 \ge 0 \implies x \ge -11$$

• For \(y = \sqrt{x - 11} + 5\):

$$x - 11 \ge 0 \implies x \ge 11$$

• For \(y = \sqrt{x + 5} - 11\):

$$x + 5 \ge 0 \implies x \ge -5$$

• For \(y = \sqrt{x + 5} + 11\):

$$x + 5 \ge 0 \implies x \ge -5$$

Identify the matching function

The function with the domain \(x \ge -11\) is:

$$y = \sqrt{x + 11} + 5$$

Answer:

• (A) \(y = \sqrt{x + 11} + 5\) (Correct answer)
• (B) \(y = \sqrt{x - 11} + 5\)
• (C) \(y = \sqrt{x + 5} - 11\)
• (D) \(y = \sqrt{x + 5} + 11\)