Sure, let's analyze the function [tex]\( f(x) = \sqrt{-x} \)[/tex] along with other functions given in the options to determine the correct statement.
### 1. Domain and Range of [tex]\( f(x) = \sqrt{-x} \)[/tex]
- Domain: The expression inside the square root, [tex]\(-x\)[/tex], must be non-negative for [tex]\( \sqrt{-x} \)[/tex] to be defined.
- This means [tex]\( -x \geq 0 \)[/tex] or [tex]\( x \leq 0 \)[/tex].
- Domain: [tex]\( (-\infty, 0] \)[/tex]
- Range: The square root of a non-negative number is non-negative.
- So, [tex]\( \sqrt{-x} \)[/tex] produces non-negative results.
- Range: [tex]\( [0, \infty) \)[/tex]
### 2. Domain and Range of [tex]\( f(x) = -\sqrt{-x} \)[/tex]
- Domain: The condition on the domain remains the same as for [tex]\( \sqrt{-x} \)[/tex].
- [tex]\( -x \geq 0 \)[/tex] or [tex]\( x \leq 0 \)[/tex].
- Domain: [tex]\( (-\infty, 0] \)[/tex]
- Range: Multiplying the square root function by -1 makes the non-negative results into non-positive results.
- So, [tex]\( -\sqrt{-x} \)[/tex] produces non-positive results.
- Range: [tex]\( (-\infty, 0] \)[/tex]
### 3. Domain and Range of [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]
- Domain: [tex]\( [0, \infty) \)[/tex]
- Range: Multiplying the square root function by -1 changes the non-negative results to non-positive results.
- So, [tex]\( -\sqrt{x} \)[/tex] produces non-positive results.
- Range: [tex]\( (-\infty, 0] \)[/tex]
### Summary of Findings
1. The function [tex]\( f(x) = \sqrt{-x} \)[/tex] has the same domain as the function [tex]\( f(x) = -\sqrt{-x} \)[/tex].
- Both domains: [tex]\( (-\infty, 0] \)[/tex]
2. The function [tex]\( f(x) = \sqrt{-x} \)[/tex] does not have the same range as the function [tex]\( f(x) = -\sqrt{-x} \)[/tex].
- [tex]\( \sqrt{-x} \)[/tex]: Range [tex]\( [0, \infty) \)[/tex]
- [tex]\( -\sqrt{-x} \)[/tex]: Range [tex]\( (-\infty, 0] \)[/tex]
3. The function [tex]\( f(x) = \sqrt{-x} \)[/tex] does not have the same domain as the function [tex]\( f(x) = -\sqrt{x} \)[/tex].
- [tex]\( \sqrt{-x} \)[/tex]: Domain [tex]\( (-\infty, 0] \)[/tex]
- [tex]\( -\sqrt{x} \)[/tex]: Domain [tex]\( [0, \infty) \)[/tex]
4. The function [tex]\( f(x) = \sqrt{-x} \)[/tex] does not have the same range as the function [tex]\( f(x) = -\sqrt{x} \)[/tex].
- [tex]\( \sqrt{-x} \)[/tex]: Range [tex]\( [0, \infty) \)[/tex]
- [tex]\( -\sqrt{x} \)[/tex]: Range [tex]\( (-\infty, 0] \)[/tex]
Based on this detailed analysis, the true statement is:
The function [tex]\( f(x) = \sqrt{-x} \)[/tex] has the same domain as the function [tex]\( f(x) = -\sqrt{-x} \)[/tex].
