At IDNLearn.com, find answers to your most pressing questions from experts and enthusiasts alike. Discover comprehensive answers from knowledgeable members of our community, covering a wide range of topics to meet all your informational needs.
Sagot :
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]
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. Find the answers you need at IDNLearn.com. Thanks for stopping by, and come back soon for more valuable insights.
