Let's analyze each of the given functions step-by-step to determine if their range includes -4:
1. Function: [tex]\( y = \sqrt{x} - 5 \)[/tex]
To check if -4 is in the range:
[tex]\[
\begin{aligned}
y = -4 & \implies -4 = \sqrt{x} - 5 \\
\implies -4 + 5 & = \sqrt{x} \\
\implies 1 & = \sqrt{x} \\
\implies x & = 1
\end{aligned}
\][/tex]
Since [tex]\( x = 1 \)[/tex] is a valid solution (and [tex]\( x \)[/tex] must be non-negative as it is under a square root), -4 is in the range of [tex]\( y = \sqrt{x} - 5 \)[/tex].
2. Function: [tex]\( y = \sqrt{x} + 5 \)[/tex]
To check if -4 is in the range:
[tex]\[
\begin{aligned}
y = -4 & \implies -4 = \sqrt{x} + 5 \\
\implies -4 - 5 & = \sqrt{x} \\
\implies -9 & = \sqrt{x}
\end{aligned}
\][/tex]
Since the square root of a number cannot be negative, -4 is not in the range of [tex]\( y = \sqrt{x} + 5 \)[/tex].
3. Function: [tex]\( y = \sqrt{x + 5} \)[/tex]
To check if -4 is in the range:
[tex]\[
\begin{aligned}
y = -4 & \implies -4 = \sqrt{x + 5}
\end{aligned}
\][/tex]
The square root function [tex]\(\sqrt{\cdot}\)[/tex] only produces non-negative values. Therefore, [tex]\(-4\)[/tex] cannot be the output of a square root function. Consequently, -4 is not in the range of [tex]\( y = \sqrt{x + 5} \)[/tex].
4. Function: [tex]\( y = \sqrt{x - 5} \)[/tex]
To check if -4 is in the range:
[tex]\[
\begin{aligned}
y = -4 & \implies -4 = \sqrt{x - 5}
\end{aligned}
\][/tex]
Similar to the previous case, the square root function [tex]\(\sqrt{\cdot}\)[/tex] only produces non-negative values. Therefore, [tex]\(-4\)[/tex] cannot be the output of a square root function. Thus -4 is not in the range of [tex]\( y = \sqrt{x - 5} \)[/tex].
In conclusion, the range of the function [tex]\( y = \sqrt{x} - 5 \)[/tex] includes -4.
