Let's systematically analyze each of the given functions to determine which one can have a range that includes [tex]\(-4\)[/tex].
1. Function: [tex]\( y = \sqrt{x} - 5 \)[/tex]
Let's see if it is possible for [tex]\( y = -4 \)[/tex]:
[tex]\[
-4 = \sqrt{x} - 5
\][/tex]
By adding 5 to both sides, we obtain:
[tex]\[
1 = \sqrt{x}
\][/tex]
Squaring both sides to eliminate the square root, we get:
[tex]\[
x = 1
\][/tex]
Since [tex]\( x = 1 \)[/tex] is a valid value (it's non-negative), [tex]\( y = \sqrt{x} - 5 \)[/tex] can indeed equal [tex]\(-4\)[/tex] when [tex]\( x = 1 \)[/tex].
Thus, the range of [tex]\( y = \sqrt{x} - 5 \)[/tex] includes [tex]\(-4\)[/tex].
2. Function: [tex]\( y = \sqrt{x} + 5 \)[/tex]
Similarly, we try to see if [tex]\( y = -4 \)[/tex] is possible:
[tex]\[
-4 = \sqrt{x} + 5
\][/tex]
Subtracting 5 from both sides, we find:
[tex]\[
-9 = \sqrt{x}
\][/tex]
This result is impossible because the square root of [tex]\( x \)[/tex] (where [tex]\( x \geq 0 \)[/tex]) is always non-negative. Thus, [tex]\( y = \sqrt{x} + 5 \)[/tex] cannot have a value of [tex]\(-4\)[/tex].
3. Function: [tex]\( y = \sqrt{x+5} \)[/tex]
Check if [tex]\( y = -4 \)[/tex]:
[tex]\[
-4 = \sqrt{x+5}
\][/tex]
Similar to the previous case, squaring both sides gives:
[tex]\[
16 = x + 5
\][/tex]
Solving for [tex]\( x \)[/tex], we get:
[tex]\[
x = 16 - 5 = 11
\][/tex]
However, when we checked this previously, squaring gave an incorrect condition since square roots do not yield negative numbers.
In this particular circumstance, [tex]\( y = \sqrt{x+5} \)[/tex] cannot equal [tex]\(-4\)[/tex].
4. Function: [tex]\( y = \sqrt{x-5} \)[/tex]
Let's investigate if [tex]\( y = -4 \)[/tex]:
[tex]\[
-4 = \sqrt{x-5}
\][/tex]
Squaring both sides results in:
[tex]\[
16 = x - 5
\][/tex]
Solving for [tex]\( x \)[/tex], we get:
[tex]\[
x = 16 + 5 = 21
\][/tex]
Once again, this situation results from incorrectly calculating negative roots.
Therefore, [tex]\( y = \sqrt{x-5} \)[/tex] also cannot equal [tex]\(-4\)[/tex].
From this detailed examination, we conclude that the range of the function [tex]\( y = \sqrt{x} - 5 \)[/tex] includes [tex]\(-4\)[/tex].
