Answer :
To determine which statement is true about the function [tex]\( f(x)=\sqrt{-x} \)[/tex], we need to analyze its domain and range and compare them with the domains and ranges of the given functions. Let's break it down step-by-step:
### Function [tex]\( f(x) = \sqrt{-x} \)[/tex]
- Domain: The expression inside the square root, [tex]\(-x\)[/tex], must be non-negative for the square root to be defined. Therefore:
[tex]\[ -x \geq 0 \implies x \leq 0 \][/tex]
So, the domain of [tex]\( f(x) = \sqrt{-x} \)[/tex] is [tex]\( (-\infty, 0] \)[/tex].
- Range: The square root function returns non-negative values. Thus:
[tex]\[ \sqrt{-x} \geq 0 \][/tex]
The range of [tex]\( f(x) = \sqrt{-x} \)[/tex] is [tex]\( [0, \infty) \)[/tex].
### Function [tex]\( f(x) = -\sqrt{-x} \)[/tex]
- Domain: Similarly, the expression inside the square root, [tex]\(-x\)[/tex], must be non-negative:
[tex]\[ -x \geq 0 \implies x \leq 0 \][/tex]
So, the domain of [tex]\( f(x) = -\sqrt{-x} \)[/tex] is [tex]\( (-\infty, 0] \)[/tex].
- Range: The negative sign in front of the square root will always return non-positive values:
[tex]\[ -\sqrt{-x} \leq 0 \][/tex]
The range of [tex]\( f(x) = -\sqrt{-x} \)[/tex] is [tex]\( (-\infty, 0] \)[/tex].
### Function [tex]\( f(x) = -\sqrt{x} \)[/tex]
- Domain: The expression inside the square root, [tex]\(x\)[/tex], must be non-negative:
[tex]\[ x \geq 0 \][/tex]
So, the domain of [tex]\( f(x) = -\sqrt{x} \)[/tex] is [tex]\( [0, \infty) \)[/tex].
- Range: The negative sign in front of the square root will always return non-positive values:
[tex]\[ -\sqrt{x} \leq 0 \][/tex]
The range of [tex]\( f(x) = -\sqrt{x} \)[/tex] is [tex]\( (-\infty, 0] \)[/tex].
### Evaluating the Statements
Given the above analysis, let's evaluate each statement:
1. It has the same domain as the function [tex]\( f(x)=-\sqrt{-x} \)[/tex].
- [tex]\( f(x) = \sqrt{-x} \)[/tex] has the domain [tex]\( (-\infty, 0] \)[/tex].
- [tex]\( f(x) = -\sqrt{-x} \)[/tex] also has the domain [tex]\( (-\infty, 0] \)[/tex].
- Therefore, this statement is true.
2. It has the same range as the function [tex]\( f(x)=-\sqrt{-x} \)[/tex].
- [tex]\( f(x) = \sqrt{-x} \)[/tex] has the range [tex]\( [0, \infty) \)[/tex].
- [tex]\( f(x) = -\sqrt{-x} \)[/tex] has the range [tex]\( (-\infty, 0] \)[/tex].
- These ranges are different, so this statement is false.
3. It has the same domain as the function [tex]\( f(x)=-\sqrt{x} \)[/tex].
- [tex]\( f(x) = \sqrt{-x} \)[/tex] has the domain [tex]\( (-\infty, 0] \)[/tex].
- [tex]\( f(x) = -\sqrt{x} \)[/tex] has the domain [tex]\( [0, \infty) \)[/tex].
- These domains are different, so this statement is false.
4. It has the same range as the function [tex]\( f(x)=-\sqrt{x} \)[/tex].
- [tex]\( f(x) = \sqrt{-x} \)[/tex] has the range [tex]\( [0, \infty) \)[/tex].
- [tex]\( f(x) = -\sqrt{x} \)[/tex] has the range [tex]\( (-\infty, 0] \)[/tex].
- These ranges are different, so this statement is false.
### Conclusion
The correct statement is:
- It has the same domain as the function [tex]\( f(x)=-\sqrt{-x} \)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{1} \][/tex]
### Function [tex]\( f(x) = \sqrt{-x} \)[/tex]
- Domain: The expression inside the square root, [tex]\(-x\)[/tex], must be non-negative for the square root to be defined. Therefore:
[tex]\[ -x \geq 0 \implies x \leq 0 \][/tex]
So, the domain of [tex]\( f(x) = \sqrt{-x} \)[/tex] is [tex]\( (-\infty, 0] \)[/tex].
- Range: The square root function returns non-negative values. Thus:
[tex]\[ \sqrt{-x} \geq 0 \][/tex]
The range of [tex]\( f(x) = \sqrt{-x} \)[/tex] is [tex]\( [0, \infty) \)[/tex].
### Function [tex]\( f(x) = -\sqrt{-x} \)[/tex]
- Domain: Similarly, the expression inside the square root, [tex]\(-x\)[/tex], must be non-negative:
[tex]\[ -x \geq 0 \implies x \leq 0 \][/tex]
So, the domain of [tex]\( f(x) = -\sqrt{-x} \)[/tex] is [tex]\( (-\infty, 0] \)[/tex].
- Range: The negative sign in front of the square root will always return non-positive values:
[tex]\[ -\sqrt{-x} \leq 0 \][/tex]
The range of [tex]\( f(x) = -\sqrt{-x} \)[/tex] is [tex]\( (-\infty, 0] \)[/tex].
### Function [tex]\( f(x) = -\sqrt{x} \)[/tex]
- Domain: The expression inside the square root, [tex]\(x\)[/tex], must be non-negative:
[tex]\[ x \geq 0 \][/tex]
So, the domain of [tex]\( f(x) = -\sqrt{x} \)[/tex] is [tex]\( [0, \infty) \)[/tex].
- Range: The negative sign in front of the square root will always return non-positive values:
[tex]\[ -\sqrt{x} \leq 0 \][/tex]
The range of [tex]\( f(x) = -\sqrt{x} \)[/tex] is [tex]\( (-\infty, 0] \)[/tex].
### Evaluating the Statements
Given the above analysis, let's evaluate each statement:
1. It has the same domain as the function [tex]\( f(x)=-\sqrt{-x} \)[/tex].
- [tex]\( f(x) = \sqrt{-x} \)[/tex] has the domain [tex]\( (-\infty, 0] \)[/tex].
- [tex]\( f(x) = -\sqrt{-x} \)[/tex] also has the domain [tex]\( (-\infty, 0] \)[/tex].
- Therefore, this statement is true.
2. It has the same range as the function [tex]\( f(x)=-\sqrt{-x} \)[/tex].
- [tex]\( f(x) = \sqrt{-x} \)[/tex] has the range [tex]\( [0, \infty) \)[/tex].
- [tex]\( f(x) = -\sqrt{-x} \)[/tex] has the range [tex]\( (-\infty, 0] \)[/tex].
- These ranges are different, so this statement is false.
3. It has the same domain as the function [tex]\( f(x)=-\sqrt{x} \)[/tex].
- [tex]\( f(x) = \sqrt{-x} \)[/tex] has the domain [tex]\( (-\infty, 0] \)[/tex].
- [tex]\( f(x) = -\sqrt{x} \)[/tex] has the domain [tex]\( [0, \infty) \)[/tex].
- These domains are different, so this statement is false.
4. It has the same range as the function [tex]\( f(x)=-\sqrt{x} \)[/tex].
- [tex]\( f(x) = \sqrt{-x} \)[/tex] has the range [tex]\( [0, \infty) \)[/tex].
- [tex]\( f(x) = -\sqrt{x} \)[/tex] has the range [tex]\( (-\infty, 0] \)[/tex].
- These ranges are different, so this statement is false.
### Conclusion
The correct statement is:
- It has the same domain as the function [tex]\( f(x)=-\sqrt{-x} \)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{1} \][/tex]
