Let's analyze the function [tex]\( f(x) = \sqrt{-x} \)[/tex] step-by-step to determine the correct statements about its domain and range.
### Domain:
1. The function [tex]\( f(x) = \sqrt{-x} \)[/tex] involves taking the square root of [tex]\(-x\)[/tex].
2. For the square root function to output real numbers, the expression inside the square root must be non-negative (i.e., [tex]\(\sqrt{y}\)[/tex] is defined for [tex]\( y \geq 0 \)[/tex]).
3. Therefore, we need [tex]\(-x \geq 0\)[/tex].
To solve [tex]\( -x \geq 0 \)[/tex]:
[tex]\[ -x \geq 0 \][/tex]
[tex]\[ x \leq 0 \][/tex]
Thus, the domain of the function [tex]\( f(x) \)[/tex] is all real numbers [tex]\( x \)[/tex] such that [tex]\( x \leq 0 \)[/tex].
### Range:
1. The square root function [tex]\( \sqrt{y} \)[/tex] always produces non-negative results since the square root of a non-negative number [tex]\( y \geq 0 \)[/tex] is also non-negative.
2. Let's consider the output of [tex]\( f(x) = \sqrt{-x} \)[/tex] as [tex]\( x \)[/tex] varies over its domain [tex]\( x \leq 0 \)[/tex].
3. When [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \sqrt{-0} = \sqrt{0} = 0 \][/tex]
4. As [tex]\( x \)[/tex] decreases (becomes more negative), [tex]\(-x\)[/tex] increases (becomes more positive), and thus [tex]\( f(x) \)[/tex] increases.
5. Therefore, for every [tex]\( x \leq 0 \)[/tex]:
[tex]\[ f(x) = \sqrt{-x} \geq 0 \][/tex]
Thus, the range of the function [tex]\( f(x) \)[/tex] is all real numbers [tex]\( y \)[/tex] such that [tex]\( y \geq 0 \)[/tex].
### Conclusion:
Based on the analysis:
- The domain of the function [tex]\( f(x) = \sqrt{-x} \)[/tex] is all real numbers less than or equal to 0.
- The range of the function [tex]\( f(x) = \sqrt{-x} \)[/tex] is all real numbers greater than or equal to 0.
The true statement among the choices given is:
- The domain of the graph is all real numbers less than or equal to 0.
So the correct statement is:
The domain of the graph is all real numbers less than or equal to 0.
