To determine the range of the function [tex]\( y = \sqrt{x+5} \)[/tex], let's analyze it step by step.
1. Understand the Function: The function [tex]\( y = \sqrt{x+5} \)[/tex] is defined for all [tex]\( x \)[/tex] such that the expression inside the square root is non-negative, because the square root of a negative number is not a real number.
2. Domain of the Function: For [tex]\( y = \sqrt{x+5} \)[/tex] to be real, [tex]\( x+5 \)[/tex] must be greater than or equal to zero:
[tex]\[
x + 5 \geq 0 \implies x \geq -5
\][/tex]
Therefore, the domain of the function is [tex]\( x \geq -5 \)[/tex].
3. Minimum Value of [tex]\( x \)[/tex]: The smallest value that [tex]\( x \)[/tex] can take is [tex]\( -5 \)[/tex].
4. Evaluate [tex]\( y \)[/tex] for this Minimum [tex]\( x \)[/tex]:
[tex]\[
y = \sqrt{x+5}
\][/tex]
Substituting the minimum value [tex]\( x = -5 \)[/tex]:
[tex]\[
y = \sqrt{-5 + 5} = \sqrt{0} = 0
\][/tex]
Hence, the minimum value of [tex]\( y \)[/tex] is 0.
5. Positive Nature of Square Root: The square root function only produces non-negative results. Therefore, for any [tex]\( x \geq -5 \)[/tex], [tex]\( y \)[/tex] will be at least 0 and can take any value greater than or equal to 0.
6. Conclusion: The range of the function [tex]\( y = \sqrt{x+5} \)[/tex] is all real numbers [tex]\( y \)[/tex] such that [tex]\( y \geq 0 \)[/tex].
Thus, the correct answer is:
[tex]\[ y \geq 0 \][/tex]
