To determine which function among the given options has the domain [tex]\( x \geq -11 \)[/tex], let's analyze each function individually.
1. Function: [tex]\( y = \sqrt{x + 11} + 5 \)[/tex]
- The argument inside the square root, [tex]\( x + 11 \)[/tex], must be non-negative for the function to be defined because the square root function is only defined for non-negative numbers.
- Therefore, [tex]\( x + 11 \geq 0 \)[/tex]
- Rearrange to solve for [tex]\( x \)[/tex]:
[tex]\[
x \geq -11
\][/tex]
- This means the domain of [tex]\( y = \sqrt{x + 11} + 5 \)[/tex] is [tex]\( x \geq -11 \)[/tex].
2. Function: [tex]\( y = \sqrt{x - 11} + 5 \)[/tex]
- The argument inside the square root, [tex]\( x - 11 \)[/tex], must be non-negative:
[tex]\[
x - 11 \geq 0
\][/tex]
- Rearrange to solve for [tex]\( x \)[/tex]:
[tex]\[
x \geq 11
\][/tex]
- The domain of [tex]\( y = \sqrt{x - 11} + 5 \)[/tex] is [tex]\( x \geq 11 \)[/tex], which is different from [tex]\( x \geq -11 \)[/tex].
3. Function: [tex]\( y = \sqrt{x + 5} - 11 \)[/tex]
- The argument inside the square root, [tex]\( x + 5 \)[/tex], must be non-negative:
[tex]\[
x + 5 \geq 0
\][/tex]
- Rearrange to solve for [tex]\( x \)[/tex]:
[tex]\[
x \geq -5
\][/tex]
- The domain of [tex]\( y = \sqrt{x + 5} - 11 \)[/tex] is [tex]\( x \geq -5 \)[/tex], which is not the same as [tex]\( x \geq -11 \)[/tex].
4. Function: [tex]\( y = \sqrt{x + 5} + 1 - 1 \)[/tex]
- Simplifying the function, we get:
[tex]\[
y = \sqrt{x + 5}
\][/tex]
- The argument inside the square root, [tex]\( x + 5 \)[/tex], must be non-negative:
[tex]\[
x + 5 \geq 0
\][/tex]
- Rearrange to solve for [tex]\( x \)[/tex]:
[tex]\[
x \geq -5
\][/tex]
- The domain of [tex]\( y = \sqrt{x + 5} + 1 - 1 \)[/tex] is [tex]\( x \geq -5 \)[/tex], which is not the same as [tex]\( x \geq -11 \)[/tex].
Conclusively, the function among the given options that has the domain [tex]\( x \geq -11 \)[/tex] is:
[tex]\[ y = \sqrt{x + 11} + 5 \][/tex]
