To determine the function with the domain [tex]\( x \geq -11 \)[/tex], we need to examine each function and identify the constraints on [tex]\( x \)[/tex] for the expression under the square root to be non-negative. This is because the square root function is only defined for non-negative values. Let's analyze each function step-by-step:
1. Function: [tex]\( y = \sqrt{x + 11} + 5 \)[/tex]
- The expression inside the square root is [tex]\( x + 11 \)[/tex].
- For the square root to be defined, [tex]\( x + 11 \geq 0 \)[/tex].
- Solving for [tex]\( x \)[/tex]:
[tex]\[
x + 11 \geq 0 \implies x \geq -11
\][/tex]
- The domain for this function is [tex]\( x \geq -11 \)[/tex].
2. Function: [tex]\( y = \sqrt{x - 11} + 5 \)[/tex]
- The expression inside the square root is [tex]\( x - 11 \)[/tex].
- For the square root to be defined, [tex]\( x - 11 \geq 0 \)[/tex].
- Solving for [tex]\( x \)[/tex]:
[tex]\[
x - 11 \geq 0 \implies x \geq 11
\][/tex]
- The domain for this function is [tex]\( x \geq 11 \)[/tex].
3. Function: [tex]\( y = \sqrt{x + 5} - 11 \)[/tex]
- The expression inside the square root is [tex]\( x + 5 \)[/tex].
- For the square root to be defined, [tex]\( x + 5 \geq 0 \)[/tex].
- Solving for [tex]\( x \)[/tex]:
[tex]\[
x + 5 \geq 0 \implies x \geq -5
\][/tex]
- The domain for this function is [tex]\( x \geq -5 \)[/tex].
4. Function: [tex]\( y = \sqrt{x + 5} + 11 \)[/tex]
- The expression inside the square root is [tex]\( x + 5 \)[/tex].
- For the square root to be defined, [tex]\( x + 5 \geq 0 \)[/tex].
- Solving for [tex]\( x \)[/tex]:
[tex]\[
x + 5 \geq 0 \implies x \geq -5
\][/tex]
- The domain for this function is [tex]\( x \geq -5 \)[/tex].
After analyzing all the functions, we see that the function [tex]\( y = \sqrt{x + 11} + 5 \)[/tex] has the domain [tex]\( x \geq -11 \)[/tex]. Thus, the function with the domain [tex]\( x \geq -11 \)[/tex] is:
[tex]\[ y = \sqrt{x + 11} + 5 \][/tex]
