Answer :
To determine which function includes [tex]\(-4\)[/tex] in its range, we need to analyze each of the given functions one by one.
### Function 1: [tex]\( y = \sqrt{x} - 5 \)[/tex]
Let's set the function equal to [tex]\(-4\)[/tex]:
[tex]\[ \sqrt{x} - 5 = -4 \][/tex]
Add 5 to both sides:
[tex]\[ \sqrt{x} = 1 \][/tex]
Square both sides to remove the square root:
[tex]\[ x = 1 \][/tex]
Since [tex]\(x = 1\)[/tex] is a non-negative number, it is a valid solution. Therefore, [tex]\(-4\)[/tex] is included in the range of [tex]\( y = \sqrt{x} - 5 \)[/tex].
### Function 2: [tex]\( y = \sqrt{x} + 5 \)[/tex]
Set the function equal to [tex]\(-4\)[/tex]:
[tex]\[ \sqrt{x} + 5 = -4 \][/tex]
Subtract 5 from both sides:
[tex]\[ \sqrt{x} = -9 \][/tex]
Since the square root of a non-negative number cannot be negative, this equation has no real solutions. Therefore, [tex]\(-4\)[/tex] is not included in the range of [tex]\( y = \sqrt{x} + 5 \)[/tex].
### Function 3: [tex]\( y = \sqrt{x+5} \)[/tex]
Set the function equal to [tex]\(-4\)[/tex]:
[tex]\[ \sqrt{x+5} = -4 \][/tex]
The square root of a non-negative number cannot be negative. Therefore, this equation has no real solutions, and [tex]\(-4\)[/tex] is not included in the range of [tex]\( y = \sqrt{x+5} \)[/tex].
### Function 4: [tex]\( y = \sqrt{x-5} \)[/tex]
Set the function equal to [tex]\(-4\)[/tex]:
[tex]\[ \sqrt{x-5} = -4 \][/tex]
Similarly, the square root of a non-negative number cannot be negative. Therefore, this equation has no real solutions, and [tex]\(-4\)[/tex] is not included in the range of [tex]\( y = \sqrt{x-5} \)[/tex].
### Conclusion:
The only function whose range includes [tex]\(-4\)[/tex] is:
[tex]\[ y = \sqrt{x} - 5 \][/tex]
### Function 1: [tex]\( y = \sqrt{x} - 5 \)[/tex]
Let's set the function equal to [tex]\(-4\)[/tex]:
[tex]\[ \sqrt{x} - 5 = -4 \][/tex]
Add 5 to both sides:
[tex]\[ \sqrt{x} = 1 \][/tex]
Square both sides to remove the square root:
[tex]\[ x = 1 \][/tex]
Since [tex]\(x = 1\)[/tex] is a non-negative number, it is a valid solution. Therefore, [tex]\(-4\)[/tex] is included in the range of [tex]\( y = \sqrt{x} - 5 \)[/tex].
### Function 2: [tex]\( y = \sqrt{x} + 5 \)[/tex]
Set the function equal to [tex]\(-4\)[/tex]:
[tex]\[ \sqrt{x} + 5 = -4 \][/tex]
Subtract 5 from both sides:
[tex]\[ \sqrt{x} = -9 \][/tex]
Since the square root of a non-negative number cannot be negative, this equation has no real solutions. Therefore, [tex]\(-4\)[/tex] is not included in the range of [tex]\( y = \sqrt{x} + 5 \)[/tex].
### Function 3: [tex]\( y = \sqrt{x+5} \)[/tex]
Set the function equal to [tex]\(-4\)[/tex]:
[tex]\[ \sqrt{x+5} = -4 \][/tex]
The square root of a non-negative number cannot be negative. Therefore, this equation has no real solutions, and [tex]\(-4\)[/tex] is not included in the range of [tex]\( y = \sqrt{x+5} \)[/tex].
### Function 4: [tex]\( y = \sqrt{x-5} \)[/tex]
Set the function equal to [tex]\(-4\)[/tex]:
[tex]\[ \sqrt{x-5} = -4 \][/tex]
Similarly, the square root of a non-negative number cannot be negative. Therefore, this equation has no real solutions, and [tex]\(-4\)[/tex] is not included in the range of [tex]\( y = \sqrt{x-5} \)[/tex].
### Conclusion:
The only function whose range includes [tex]\(-4\)[/tex] is:
[tex]\[ y = \sqrt{x} - 5 \][/tex]
