To find the range of the function [tex]\( y = \sqrt{x + 5} \)[/tex], we need to consider the values that [tex]\( y \)[/tex] can take.
1. Domain of the function:
- The expression inside the square root, [tex]\( x + 5 \)[/tex], must be non-negative because the square root of a negative number is not defined in the real number system.
- Therefore, the condition is [tex]\( x + 5 \geq 0 \)[/tex], which simplifies to [tex]\( x \geq -5 \)[/tex].
2. Possible values for [tex]\( y \)[/tex]:
- When [tex]\( x = -5 \)[/tex], the expression inside the square root becomes [tex]\( -5 + 5 = 0 \)[/tex].
- Thus, [tex]\( y = \sqrt{0} = 0 \)[/tex].
3. Behavior of the function:
- For all [tex]\( x \geq -5 \)[/tex], [tex]\( x + 5 \)[/tex] is non-negative, and the square root of a non-negative number is always non-negative.
- Thus, [tex]\( \sqrt{x + 5} \)[/tex] yields only non-negative values.
- As [tex]\( x \)[/tex] increases, [tex]\( x + 5 \)[/tex] increases, and so does [tex]\( \sqrt{x + 5} \)[/tex].
In conclusion, the function [tex]\( y = \sqrt{x + 5} \)[/tex] produces non-negative values for all permissible [tex]\( x \)[/tex]. Therefore, the range of the function is [tex]\( y \geq 0 \)[/tex].
The range of [tex]\( y = \sqrt{x + 5} \)[/tex] is:
[tex]\[ y \geq 0 \][/tex]
Hence, the correct answer is:
[tex]\[ y \geq 0 \][/tex]
