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]