Let's analyze each function step by step to determine the accurate relationships between their domains and ranges.
### Function [tex]\( f(x) = -\sqrt{x} \)[/tex]
1. Domain of [tex]\( f(x) = -\sqrt{x} \)[/tex]
- For [tex]\( \sqrt{x} \)[/tex] to be defined, [tex]\( x \geq 0 \)[/tex].
- Therefore, the domain of [tex]\( f(x) \)[/tex] is [tex]\( [0, \infty) \)[/tex].
2. Range of [tex]\( f(x) = -\sqrt{x} \)[/tex]
- Since [tex]\( f(x) \)[/tex] takes the negative of [tex]\( \sqrt{x} \)[/tex], the values will be non-positive.
- Therefore, the range of [tex]\( f(x) \)[/tex] is [tex]\( (-\infty, 0] \)[/tex].
### Function [tex]\( f(x) = \sqrt{x} \)[/tex]
1. Domain of [tex]\( f(x) = \sqrt{x} \)[/tex]
- For [tex]\( \sqrt{x} \)[/tex] to be defined, [tex]\( x \geq 0 \)[/tex].
- The domain of [tex]\( \sqrt{x} \)[/tex] is [tex]\( [0, \infty) \)[/tex].
2. Range of [tex]\( f(x) = \sqrt{x} \)[/tex]
- Since [tex]\( \sqrt{x} \)[/tex] yields non-negative values.
- Therefore, the range of [tex]\( \sqrt{x} \)[/tex] is [tex]\( [0, \infty) \)[/tex].
### Function [tex]\( f(x) = -\sqrt{-x} \)[/tex]
1. Domain of [tex]\( f(x) = -\sqrt{-x} \)[/tex]
- For [tex]\( \sqrt{-x} \)[/tex] to be defined, [tex]\( -x \geq 0 \)[/tex], which implies [tex]\( x \leq 0 \)[/tex].
- The domain of [tex]\( -\sqrt{-x} \)[/tex] is [tex]\( (-\infty, 0] \)[/tex].
2. Range of [tex]\( f(x) = -\sqrt{-x} \)[/tex]
- Since [tex]\( -\sqrt{-x} \)[/tex] yields non-positive values (similar to [tex]\( -\sqrt{x} \)[/tex]).
- Therefore, the range of [tex]\( -\sqrt{-x} \)[/tex] is [tex]\( (-\infty, 0] \)[/tex].
### Comparisons
- Comparing [tex]\( f(x) = -\sqrt{x} \)[/tex] and [tex]\( f(x) = \sqrt{x} \)[/tex]
- Domains: Both have the domain [tex]\( [0, \infty) \)[/tex].
- Ranges: [tex]\( -\sqrt{x} \)[/tex] has the range [tex]\( (-\infty, 0] \)[/tex], while [tex]\( \sqrt{x} \)[/tex] has the range [tex]\( [0, \infty) \)[/tex].
- Conclusion: They do not have the same range.
- Comparing [tex]\( f(x) = -\sqrt{x} \)[/tex] and [tex]\( f(x) = -\sqrt{-x} \)[/tex]
- Domains: [tex]\( -\sqrt{x} \)[/tex] has the domain [tex]\( [0, \infty) \)[/tex], and [tex]\( -\sqrt{-x} \)[/tex] has the domain [tex]\( (-\infty, 0] \)[/tex].
- Ranges: Both have the range [tex]\( (-\infty, 0] \)[/tex].
- Conclusion: They do not have the same domain, but they do have the same range.
### Final Answer
The correct statement is:
- "It has the same range but not the same domain as the function [tex]\( f(x) = -\sqrt{-x} \)[/tex]."
