To determine the range of the function [tex]\( y = \sqrt{x+7} + 5 \)[/tex], follow these steps:
1. Identify the form of the function:
The function is given by [tex]\( y = \sqrt{x+7} + 5 \)[/tex].
2. Determine the domain of the function:
For the square root [tex]\( \sqrt{x+7} \)[/tex] to be defined, the expression inside the square root must be non-negative. Therefore, [tex]\( x + 7 \geq 0 \)[/tex], which simplifies to [tex]\( x \geq -7 \)[/tex]. So the function is defined for all [tex]\( x \geq -7 \)[/tex].
3. Analyze the minimum value of [tex]\( y \)[/tex]:
At the smallest value in the domain [tex]\( x = -7 \)[/tex]:
[tex]\[
y = \sqrt{-7 + 7} + 5 = \sqrt{0} + 5 = 0 + 5 = 5
\][/tex]
So, the minimum value of [tex]\( y \)[/tex] is 5 when [tex]\( x = -7 \)[/tex].
4. Consider the behavior of [tex]\( y \)[/tex] as [tex]\( x \)[/tex] increases:
As [tex]\( x \)[/tex] increases from [tex]\(-7\)[/tex] to [tex]\(\infty\)[/tex], [tex]\( x+7 \)[/tex] increases, and consequently, the square root [tex]\(\sqrt{x+7}\)[/tex] increases. Hence, [tex]\( y = \sqrt{x+7} + 5 \)[/tex] increases without bound.
Therefore, [tex]\( y \)[/tex] can be any value starting from 5 and going to infinity. Therefore, the range of [tex]\( y \)[/tex] is [tex]\( y \geq 5 \)[/tex].
Conclusion:
The correct answer is [tex]\( y \geq 5 \)[/tex]. The other choices ( [tex]\( y \geq -5 \)[/tex], [tex]\( y \geq -7 \)[/tex], and "all real numbers") are incorrect in the context of this function.
Thus, the range of [tex]\( y = \sqrt{x+7} + 5 \)[/tex] is:
[tex]\[ y \geq 5 \][/tex]
