Answer :
To determine which functions can have [tex]\(-4\)[/tex] in their range, we need to examine each function step by step. Recall that the range is the set of all possible output values [tex]\( y \)[/tex] of the function.
Let's analyze each function one by one.
1. Function: [tex]\( y = \sqrt{x} - 5 \)[/tex]
First, represent the function as:
[tex]\[ y = \sqrt{x} - 5 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ y + 5 = \sqrt{x} \][/tex]
[tex]\[ x = (y + 5)^2 \][/tex]
The domain of the square root function, [tex]\(\sqrt{x}\)[/tex], requires that:
[tex]\[ x \geq 0 \][/tex]
Thus,
[tex]\[ (y + 5)^2 \geq 0 \][/tex]
This constraint is always met since a square is always non-negative. Therefore, the range of possible [tex]\( y \)[/tex] values can be found by:
[tex]\[ \sqrt{x} \geq 0 \][/tex]
[tex]\[ \sqrt{x} - 5 \geq -5 \][/tex]
Thus:
[tex]\[ y \geq -5 \][/tex]
Conclusion: The function [tex]\( y = \sqrt{x} - 5 \)[/tex] has a range of [tex]\( y \geq -5 \)[/tex]. Therefore, it can include [tex]\(-4\)[/tex].
2. Function: [tex]\( y = \sqrt{x} + 5 \)[/tex]
First, represent the function as:
[tex]\[ y = \sqrt{x} + 5 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ y - 5 = \sqrt{x} \][/tex]
[tex]\[ x = (y - 5)^2 \][/tex]
The domain of the square root function, [tex]\(\sqrt{x}\)[/tex], requires that:
[tex]\[ x \geq 0 \][/tex]
Thus,
[tex]\[ (y - 5)^2 \geq 0 \][/tex]
This constraint is always met since a square is always non-negative. Therefore, the range of possible [tex]\( y \)[/tex] values can be found by:
[tex]\[ \sqrt{x} \geq 0 \][/tex]
[tex]\[ \sqrt{x} + 5 \geq 5 \][/tex]
Thus:
[tex]\[ y \geq 5 \][/tex]
Conclusion: The function [tex]\( y = \sqrt{x} + 5 \)[/tex] has a range of [tex]\( y \geq 5 \)[/tex]. Therefore, it cannot include [tex]\(-4\)[/tex].
3. Function: [tex]\( y = \sqrt{x + 5} \)[/tex]
First, represent the function as:
[tex]\[ y = \sqrt{x + 5} \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ y^2 = x + 5 \][/tex]
[tex]\[ x = y^2 - 5 \][/tex]
The domain of the square root function, [tex]\(\sqrt{x+5}\)[/tex], requires that:
[tex]\[ x + 5 \geq 0 \][/tex]
Thus,
[tex]\[ x \geq -5 \][/tex]
The range of possible [tex]\( y \)[/tex] values starts from the minimum value of [tex]\(\sqrt{x+5}\)[/tex], which is at [tex]\( x = -5 \)[/tex]:
[tex]\[ \sqrt{x + 5} \geq 0 \][/tex]
Thus:
[tex]\[ y \geq 0 \][/tex]
Conclusion: The function [tex]\( y = \sqrt{x + 5} \)[/tex] has a range of [tex]\( y \geq 0 \)[/tex]. Therefore, it cannot include [tex]\(-4\)[/tex].
4. Function: [tex]\( y = \sqrt{x - 5} \)[/tex]
First, represent the function as:
[tex]\[ y = \sqrt{x - 5} \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ y^2 = x - 5 \][/tex]
[tex]\[ x = y^2 + 5 \][/tex]
The domain of the square root function, [tex]\(\sqrt{x-5}\)[/tex], requires that:
[tex]\[ x - 5 \geq 0 \][/tex]
Thus,
[tex]\[ x \geq 5 \][/tex]
The range of possible [tex]\( y \)[/tex] values starts from the minimum value of [tex]\(\sqrt{x-5}\)[/tex], which is at [tex]\( x = 5 \)[/tex]:
[tex]\[ \sqrt{x - 5} \geq 0 \][/tex]
Thus:
[tex]\[ y \geq 0 \][/tex]
Conclusion: The function [tex]\( y = \sqrt{x - 5} \)[/tex] has a range of [tex]\( y \geq 0 \)[/tex]. Therefore, it cannot include [tex]\(-4\)[/tex].
### Final Answer:
Only the function [tex]\( \boxed{y = \sqrt{x} - 5 } \)[/tex] has a range that can include [tex]\(-4\)[/tex].
Let's analyze each function one by one.
1. Function: [tex]\( y = \sqrt{x} - 5 \)[/tex]
First, represent the function as:
[tex]\[ y = \sqrt{x} - 5 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ y + 5 = \sqrt{x} \][/tex]
[tex]\[ x = (y + 5)^2 \][/tex]
The domain of the square root function, [tex]\(\sqrt{x}\)[/tex], requires that:
[tex]\[ x \geq 0 \][/tex]
Thus,
[tex]\[ (y + 5)^2 \geq 0 \][/tex]
This constraint is always met since a square is always non-negative. Therefore, the range of possible [tex]\( y \)[/tex] values can be found by:
[tex]\[ \sqrt{x} \geq 0 \][/tex]
[tex]\[ \sqrt{x} - 5 \geq -5 \][/tex]
Thus:
[tex]\[ y \geq -5 \][/tex]
Conclusion: The function [tex]\( y = \sqrt{x} - 5 \)[/tex] has a range of [tex]\( y \geq -5 \)[/tex]. Therefore, it can include [tex]\(-4\)[/tex].
2. Function: [tex]\( y = \sqrt{x} + 5 \)[/tex]
First, represent the function as:
[tex]\[ y = \sqrt{x} + 5 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ y - 5 = \sqrt{x} \][/tex]
[tex]\[ x = (y - 5)^2 \][/tex]
The domain of the square root function, [tex]\(\sqrt{x}\)[/tex], requires that:
[tex]\[ x \geq 0 \][/tex]
Thus,
[tex]\[ (y - 5)^2 \geq 0 \][/tex]
This constraint is always met since a square is always non-negative. Therefore, the range of possible [tex]\( y \)[/tex] values can be found by:
[tex]\[ \sqrt{x} \geq 0 \][/tex]
[tex]\[ \sqrt{x} + 5 \geq 5 \][/tex]
Thus:
[tex]\[ y \geq 5 \][/tex]
Conclusion: The function [tex]\( y = \sqrt{x} + 5 \)[/tex] has a range of [tex]\( y \geq 5 \)[/tex]. Therefore, it cannot include [tex]\(-4\)[/tex].
3. Function: [tex]\( y = \sqrt{x + 5} \)[/tex]
First, represent the function as:
[tex]\[ y = \sqrt{x + 5} \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ y^2 = x + 5 \][/tex]
[tex]\[ x = y^2 - 5 \][/tex]
The domain of the square root function, [tex]\(\sqrt{x+5}\)[/tex], requires that:
[tex]\[ x + 5 \geq 0 \][/tex]
Thus,
[tex]\[ x \geq -5 \][/tex]
The range of possible [tex]\( y \)[/tex] values starts from the minimum value of [tex]\(\sqrt{x+5}\)[/tex], which is at [tex]\( x = -5 \)[/tex]:
[tex]\[ \sqrt{x + 5} \geq 0 \][/tex]
Thus:
[tex]\[ y \geq 0 \][/tex]
Conclusion: The function [tex]\( y = \sqrt{x + 5} \)[/tex] has a range of [tex]\( y \geq 0 \)[/tex]. Therefore, it cannot include [tex]\(-4\)[/tex].
4. Function: [tex]\( y = \sqrt{x - 5} \)[/tex]
First, represent the function as:
[tex]\[ y = \sqrt{x - 5} \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ y^2 = x - 5 \][/tex]
[tex]\[ x = y^2 + 5 \][/tex]
The domain of the square root function, [tex]\(\sqrt{x-5}\)[/tex], requires that:
[tex]\[ x - 5 \geq 0 \][/tex]
Thus,
[tex]\[ x \geq 5 \][/tex]
The range of possible [tex]\( y \)[/tex] values starts from the minimum value of [tex]\(\sqrt{x-5}\)[/tex], which is at [tex]\( x = 5 \)[/tex]:
[tex]\[ \sqrt{x - 5} \geq 0 \][/tex]
Thus:
[tex]\[ y \geq 0 \][/tex]
Conclusion: The function [tex]\( y = \sqrt{x - 5} \)[/tex] has a range of [tex]\( y \geq 0 \)[/tex]. Therefore, it cannot include [tex]\(-4\)[/tex].
### Final Answer:
Only the function [tex]\( \boxed{y = \sqrt{x} - 5 } \)[/tex] has a range that can include [tex]\(-4\)[/tex].
