To determine the range of the function [tex]\( y = \sqrt{x+5} \)[/tex], let's analyze its behavior step-by-step.
1. Identify the Domain of the Function:
The function [tex]\( y = \sqrt{x+5} \)[/tex] involves a square root. The expression inside the square root, [tex]\( x+5 \)[/tex], must be non-negative because the square root of a negative number is not a real number.
Thus, we have:
[tex]\[
x + 5 \geq 0
\][/tex]
Simplifying this inequality, we get:
[tex]\[
x \geq -5
\][/tex]
So, the domain of the function is [tex]\( x \geq -5 \)[/tex].
2. Evaluate the Minimum Value of the Function:
Next, we need to determine the minimum value of [tex]\( y \)[/tex].
[tex]\[
y = \sqrt{x+5}
\][/tex]
The minimum value of the expression [tex]\( x + 5 \)[/tex] occurs when [tex]\( x \)[/tex] is at its smallest within the domain. This happens when [tex]\( x = -5 \)[/tex]:
[tex]\[
y = \sqrt{-5 + 5} = \sqrt{0} = 0
\][/tex]
Thus, the minimum value of [tex]\( y \)[/tex] is 0.
3. Determine the Range of the Function:
Since the square root function produces only non-negative numbers and we identified that the smallest value of [tex]\( y \)[/tex] is 0, the range of the function is all values [tex]\( y \)[/tex] where [tex]\( y \geq 0 \)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{y \geq 0}
\][/tex]
