Let's determine the range of the function [tex]\( y = \sqrt{x + 5} \)[/tex] step-by-step:
1. Identify the domain of the function [tex]\( y = \sqrt{x + 5} \)[/tex]:
- The expression inside the square root, [tex]\( x + 5 \)[/tex], must be greater than or equal to zero because the square root of a negative number is not defined in the set of real numbers.
- Thus, [tex]\( x + 5 \geq 0 \)[/tex].
- Solving this inequality: [tex]\( x \geq -5 \)[/tex].
2. Find the corresponding [tex]\( y \)[/tex]-values:
- The function [tex]\( y = \sqrt{x + 5} \)[/tex] takes the form of a square root, which means the output (the [tex]\( y \)[/tex]-value) is always non-negative. The square root function returns values that are zero or positive.
- When [tex]\( x = -5 \)[/tex], the smallest value in the domain, we substitute [tex]\( x = -5 \)[/tex] into the function: [tex]\( y = \sqrt{-5 + 5} = \sqrt{0} = 0 \)[/tex].
3. Determine the behavior of [tex]\( y \)[/tex] as [tex]\( x \)[/tex] increases:
- As [tex]\( x \)[/tex] increases from [tex]\(-5\)[/tex] to larger values, [tex]\( x + 5 \)[/tex] becomes larger.
- The square root of a larger positive number continues to increase, so [tex]\( y = \sqrt{x + 5} \)[/tex] becomes larger.
4. Conclude the range:
- Since the smallest value of [tex]\( y \)[/tex] is 0 (occurring when [tex]\( x = -5 \)[/tex]), and [tex]\( y \)[/tex] can take any non-negative value as [tex]\( x \)[/tex] increases, the range of [tex]\( y = \sqrt{x + 5} \)[/tex] is all non-negative real numbers.
Therefore, the correct answer is:
[tex]\[ y \geq 0 \][/tex]
