Let's analyze the function [tex]\( f(x) = \sqrt{-x} \)[/tex].
1. Domain:
- The function involves a square root, which requires its argument to be non-negative, because the square root of a negative number is not defined in the real number system.
- The argument inside the square root is [tex]\(-x\)[/tex]. Therefore, we need [tex]\(-x \geq 0\)[/tex].
- Solving this inequality, we get:
[tex]\[
-x \geq 0 \implies x \leq 0
\][/tex]
- Hence, the domain of the function is all real numbers less than or equal to 0.
2. Range:
- For the range, consider the values that [tex]\( f(x) \)[/tex] can take as [tex]\( x \)[/tex] varies over its domain.
- Since the square root function always returns non-negative values (i.e., the result is always greater than or equal to 0), we look at how [tex]\( f(x) = \sqrt{-x} \)[/tex] behaves:
- When [tex]\( x = 0 \)[/tex], [tex]\( f(0) = \sqrt{-0} = \sqrt{0} = 0 \)[/tex].
- As [tex]\( x \)[/tex] decreases (becomes more negative), [tex]\(-x\)[/tex] becomes more positive, and thus [tex]\( \sqrt{-x} \)[/tex] increases.
- Therefore, [tex]\( f(x) \)[/tex] will always produce values that are greater than or equal to 0.
- So, the range of [tex]\( f(x) \)[/tex] is all non-negative real numbers, i.e., [tex]\( [0, \infty) \)[/tex].
To summarize, among the given statements, the one that correctly describes the function [tex]\( f(x) = \sqrt{-x} \)[/tex] is:
The domain of the graph is all real numbers less than or equal to 0.
Thus, the correct statement is:
The domain of the graph is all real numbers less than or equal to 0.
