To determine which function's range includes [tex]\(-4\)[/tex], let's analyze each function separately.
1. For the function [tex]\( y = \sqrt{x} - 5 \)[/tex]:
We need to see if [tex]\( y \)[/tex] can be [tex]\(-4\)[/tex].
[tex]\[ y = \sqrt{x} - 5 = -4 \][/tex]
Adding [tex]\(5\)[/tex] to both sides:
[tex]\[ \sqrt{x} = -4 + 5 \][/tex]
[tex]\[ \sqrt{x} = 1 \][/tex]
Squaring both sides to solve for [tex]\(x\)[/tex]:
[tex]\[ x = 1 \][/tex]
Since [tex]\(x = 1\)[/tex] is a valid input (non-negative) for the square root function, we can evaluate:
[tex]\[ y = \sqrt{1} - 5 = 1 - 5 = -4 \][/tex]
Therefore, [tex]\(-4\)[/tex] is indeed in the range of the function [tex]\( y = \sqrt{x} - 5 \)[/tex].
2. For the function [tex]\( y = \sqrt{x} + 5 \)[/tex]:
We need to see if [tex]\( y \)[/tex] can be [tex]\(-4\)[/tex].
[tex]\[ y = \sqrt{x} + 5 = -4 \][/tex]
Subtracting [tex]\( 5 \)[/tex] from both sides:
[tex]\[ \sqrt{x} = -4 - 5 \][/tex]
[tex]\[ \sqrt{x} = -9 \][/tex]
Since the square root of a real number cannot be negative, there is no [tex]\( x \)[/tex] that satisfies this equation. Therefore, [tex]\(-4\)[/tex] is not in the range of [tex]\( y = \sqrt{x} + 5 \)[/tex].
3. For the function [tex]\( y = \sqrt{x+5} \)[/tex]:
We need to see if [tex]\( y \)[/tex] can be [tex]\(-4\)[/tex].
[tex]\[ y = \sqrt{x+5} = -4 \][/tex]
Squaring both sides to solve for [tex]\( x \)[/tex]:
[tex]\[ x + 5 = (-4)^2 \][/tex]
[tex]\[ x + 5 = 16 \][/tex]
Subtracting [tex]\( 5 \)[/tex] from both sides:
[tex]\[ x = 11 \][/tex]
However, this implies:
[tex]\[ y = \sqrt{x+5} = \sqrt{11 + 5} = \sqrt{16} = 4 \][/tex]
Since only positive square roots are considered (in real numbers), [tex]\(-4\)[/tex] cannot be in the range of [tex]\( y = \sqrt{x+5} \)[/tex].
4. For the function [tex]\( y = \sqrt{x - 5} \)[/tex]:
We need to see if [tex]\( y \)[/tex] can be [tex]\(-4\)[/tex].
[tex]\[ y = \sqrt{x - 5} = -4 \][/tex]
Squaring both sides to solve for [tex]\( x \)[/tex]:
[tex]\[ x - 5 = (-4)^2 \][/tex]
[tex]\[ x - 5 = 16 \][/tex]
Adding [tex]\( 5 \)[/tex] to both sides:
[tex]\[ x = 21 \][/tex]
Yet, this implies:
[tex]\[ y = \sqrt{x - 5} = \sqrt{21 - 5} = \sqrt{16} = 4 \][/tex]
Again, since only positive square roots are considered (in real numbers), [tex]\(-4\)[/tex] cannot be in the range of [tex]\( y = \sqrt{x - 5} \)[/tex].
In conclusion, the range of the function [tex]\( y = \sqrt{x} - 5 \)[/tex] includes [tex]\(-4\)[/tex].
