Question

square root functions and equations
which function has the domain $x \geq -11$?
$y = \sqrt{x + 11} + 5$
$y = \sqrt{x - 11} + 5$
$y = \sqrt{x + 5} + 11$
$y = \sqrt{x + 5} - 11$

Explanation:

Step1: Recall domain of square root

For a square root function \( y = \sqrt{f(x)} + k \), the domain requires \( f(x) \geq 0 \).

Step2: Analyze each function

• For \( y=\sqrt{x + 11}+5 \): Solve \( x + 11\geq0 \Rightarrow x\geq - 11 \).
• For \( y=\sqrt{x - 11}+5 \): Solve \( x - 11\geq0 \Rightarrow x\geq11 \).
• For \( y=\sqrt{x + 5}+11 \): Solve \( x + 5\geq0 \Rightarrow x\geq - 5 \).
• For \( y=\sqrt{x + 5}-11 \): Solve \( x + 5\geq0 \Rightarrow x\geq - 5 \).

Answer:

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