Answer :
To determine the correct statement about the function [tex]\( f(x) = -\sqrt{x} \)[/tex], we should analyze the domains and ranges of the functions mentioned:
Function [tex]\( f(x) = \sqrt{x} \)[/tex]:
1. Domain: The domain is the set of all allowable inputs for [tex]\( x \)[/tex]. Since the square root function [tex]\( \sqrt{x} \)[/tex] is only defined for non-negative numbers, [tex]\( x \)[/tex] must be greater than or equal to 0.
- Domain: [tex]\( [0, \infty) \)[/tex]
2. Range: The range is the set of all possible outputs of the function. For [tex]\( \sqrt{x} \)[/tex], since a square root function always yields non-negative results, the range is also non-negative numbers.
- Range: [tex]\( [0, \infty) \)[/tex]
Function [tex]\( f(x) = -\sqrt{x} \)[/tex]:
1. Domain: Like [tex]\( \sqrt{x} \)[/tex], the function [tex]\( -\sqrt{x} \)[/tex] is defined only for non-negative [tex]\( x \)[/tex].
- Domain: [tex]\( [0, \infty) \)[/tex]
2. Range: Since [tex]\( -\sqrt{x} \)[/tex] is the negative of the square root, the output will always be non-positive (i.e., zero or negative values).
- Range: [tex]\( (-\infty, 0] \)[/tex]
Function [tex]\( f(x) = -\sqrt{-x} \)[/tex]:
1. Domain: For [tex]\( \sqrt{-x} \)[/tex] to be defined, [tex]\( -x \)[/tex] must be non-negative. This implies [tex]\( x \)[/tex] must be non-positive.
- Domain: [tex]\( (-\infty, 0] \)[/tex]
2. Range: The output of [tex]\( -\sqrt{-x} \)[/tex] will also be non-positive since it is the negative of the square root value.
- Range: [tex]\( (-\infty, 0] \)[/tex]
Now, let's match the statements:
1. It has the same domain and range as the function [tex]\( f(x) = \sqrt{x} \)[/tex].
- This is incorrect because the range of [tex]\( f(x) = -\sqrt{x} \)[/tex] is [tex]\( (-\infty, 0] \)[/tex], which is different from the range [tex]\( [0, \infty) \)[/tex] of [tex]\( f(x) = \sqrt{x} \)[/tex].
2. It has the same range but not the same domain as the function [tex]\( f(x) = \sqrt{x} \)[/tex].
- This is incorrect since [tex]\( f(x) = -\sqrt{x} \)[/tex] does not have the same range as [tex]\( f(x) = \sqrt{x} \)[/tex].
3. It has the same domain and range as the function [tex]\( f(x) = -\sqrt{-x} \)[/tex].
- This is incorrect because the domain of [tex]\( f(x) = -\sqrt{x} \)[/tex] is [tex]\( [0, \infty) \)[/tex], which is different from the domain [tex]\( (-\infty, 0] \)[/tex] of [tex]\( f(x) = -\sqrt{-x} \)[/tex].
4. It has the same range but not the same domain as the function [tex]\( f(x) = -\sqrt{-x} \)[/tex].
- This is correct because:
- Both [tex]\( f(x) = -\sqrt{x} \)[/tex] and [tex]\( f(x) = -\sqrt{-x} \)[/tex] have the same range [tex]\( (-\infty, 0] \)[/tex].
- However, they have different domains: [tex]\( [0, \infty) \)[/tex] for [tex]\( f(x) = -\sqrt{x} \)[/tex] and [tex]\( (-\infty, 0] \)[/tex] for [tex]\( f(x) = -\sqrt{-x} \)[/tex].
Thus, the correct statement is:
It has the same range but not the same domain as the function [tex]\( f(x)= -\sqrt{-x} \)[/tex].
Function [tex]\( f(x) = \sqrt{x} \)[/tex]:
1. Domain: The domain is the set of all allowable inputs for [tex]\( x \)[/tex]. Since the square root function [tex]\( \sqrt{x} \)[/tex] is only defined for non-negative numbers, [tex]\( x \)[/tex] must be greater than or equal to 0.
- Domain: [tex]\( [0, \infty) \)[/tex]
2. Range: The range is the set of all possible outputs of the function. For [tex]\( \sqrt{x} \)[/tex], since a square root function always yields non-negative results, the range is also non-negative numbers.
- Range: [tex]\( [0, \infty) \)[/tex]
Function [tex]\( f(x) = -\sqrt{x} \)[/tex]:
1. Domain: Like [tex]\( \sqrt{x} \)[/tex], the function [tex]\( -\sqrt{x} \)[/tex] is defined only for non-negative [tex]\( x \)[/tex].
- Domain: [tex]\( [0, \infty) \)[/tex]
2. Range: Since [tex]\( -\sqrt{x} \)[/tex] is the negative of the square root, the output will always be non-positive (i.e., zero or negative values).
- Range: [tex]\( (-\infty, 0] \)[/tex]
Function [tex]\( f(x) = -\sqrt{-x} \)[/tex]:
1. Domain: For [tex]\( \sqrt{-x} \)[/tex] to be defined, [tex]\( -x \)[/tex] must be non-negative. This implies [tex]\( x \)[/tex] must be non-positive.
- Domain: [tex]\( (-\infty, 0] \)[/tex]
2. Range: The output of [tex]\( -\sqrt{-x} \)[/tex] will also be non-positive since it is the negative of the square root value.
- Range: [tex]\( (-\infty, 0] \)[/tex]
Now, let's match the statements:
1. It has the same domain and range as the function [tex]\( f(x) = \sqrt{x} \)[/tex].
- This is incorrect because the range of [tex]\( f(x) = -\sqrt{x} \)[/tex] is [tex]\( (-\infty, 0] \)[/tex], which is different from the range [tex]\( [0, \infty) \)[/tex] of [tex]\( f(x) = \sqrt{x} \)[/tex].
2. It has the same range but not the same domain as the function [tex]\( f(x) = \sqrt{x} \)[/tex].
- This is incorrect since [tex]\( f(x) = -\sqrt{x} \)[/tex] does not have the same range as [tex]\( f(x) = \sqrt{x} \)[/tex].
3. It has the same domain and range as the function [tex]\( f(x) = -\sqrt{-x} \)[/tex].
- This is incorrect because the domain of [tex]\( f(x) = -\sqrt{x} \)[/tex] is [tex]\( [0, \infty) \)[/tex], which is different from the domain [tex]\( (-\infty, 0] \)[/tex] of [tex]\( f(x) = -\sqrt{-x} \)[/tex].
4. It has the same range but not the same domain as the function [tex]\( f(x) = -\sqrt{-x} \)[/tex].
- This is correct because:
- Both [tex]\( f(x) = -\sqrt{x} \)[/tex] and [tex]\( f(x) = -\sqrt{-x} \)[/tex] have the same range [tex]\( (-\infty, 0] \)[/tex].
- However, they have different domains: [tex]\( [0, \infty) \)[/tex] for [tex]\( f(x) = -\sqrt{x} \)[/tex] and [tex]\( (-\infty, 0] \)[/tex] for [tex]\( f(x) = -\sqrt{-x} \)[/tex].
Thus, the correct statement is:
It has the same range but not the same domain as the function [tex]\( f(x)= -\sqrt{-x} \)[/tex].
