To determine the range of the function [tex]\( y = \sqrt{x+7} + 5 \)[/tex], we need to analyze the behavior of the function.
1. Identify Domain Constraints:
The square root function, [tex]\(\sqrt{x}\)[/tex], is only defined for non-negative values. This implies that for [tex]\(\sqrt{x+7}\)[/tex] to be defined, the expression inside the square root must be non-negative:
[tex]\[
x + 7 \geq 0
\][/tex]
Solving for [tex]\( x \)[/tex], we get:
[tex]\[
x \geq -7
\][/tex]
2. Determine the Minimum Value of [tex]\(y\)[/tex]:
Since [tex]\( x \geq -7 \)[/tex]:
When [tex]\( x = -7 \)[/tex]:
[tex]\[
y = \sqrt{-7 + 7} + 5 = \sqrt{0} + 5 = 0 + 5 = 5
\][/tex]
3. Behavior of the Function for [tex]\( x > -7 \)[/tex]:
As [tex]\( x \)[/tex] becomes larger than [tex]\(-7\)[/tex], the value of [tex]\( x + 7 \)[/tex] increases:
[tex]\[
\sqrt{x+7} \text{ becomes larger and positive}
\][/tex]
Hence, [tex]\( y \)[/tex] increases as well:
[tex]\[
y = \sqrt{x+7} + 5
\][/tex]
4. Conclusion on Range:
The minimum value of [tex]\( y \)[/tex] is [tex]\( 5 \)[/tex] when [tex]\( x = -7 \)[/tex]. As [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] also increases without an upper bound.
Therefore, the range of [tex]\( y \)[/tex] is:
[tex]\[
y \geq 5
\][/tex]
So the correct answer from the given choices is:
[tex]\[
\boxed{y \geq 5}
\][/tex]
