To determine which statement is true about the function [tex]\( f(x) = \sqrt{-x} \)[/tex], we need to analyze both the domain and the range of the function.
1. Understanding the Domain:
- The domain of a function is the set of all possible input values (x-values) for which the function is defined.
- For the function [tex]\( f(x) = \sqrt{-x} \)[/tex] to be defined, the expression inside the square root, which is [tex]\(-x\)[/tex], must be non-negative. This is because the square root of a negative number is not defined in the set of real numbers.
So, we need:
[tex]\[
-x \geq 0
\][/tex]
Solving this inequality for [tex]\( x \)[/tex], we get:
[tex]\[
x \leq 0
\][/tex]
Therefore, the domain of [tex]\( f(x) = \sqrt{-x} \)[/tex] is all real numbers less than or equal to 0.
2. Understanding the Range:
- The range of a function is the set of all possible output values (y-values).
- For [tex]\( f(x) = \sqrt{-x} \)[/tex], since [tex]\(-x \)[/tex] is non-negative (as we established in the domain), and the square root of a non-negative number is also non-negative, the function will only produce non-negative values.
- More specifically, if [tex]\( x \leq 0 \)[/tex], then [tex]\(-x \geq 0\)[/tex], and so the square root, [tex]\( \sqrt{-x} \)[/tex], will be greater than or equal to 0.
Therefore, the range of [tex]\( f(x) = \sqrt{-x} \)[/tex] is all real numbers greater than or equal to 0.
Given these observations, we can evaluate the statements:
1. The domain of the graph is all real numbers. - Incorrect, because the domain is limited to [tex]\( x \leq 0 \)[/tex].
2. The range of the graph is all real numbers. - Incorrect, because the range is non-negative values.
3. The domain of the graph is all real numbers less than or equal to 0. - Correct, as shown in our analysis.
4. The range of the graph is all real numbers less than or equal to 0. - Incorrect, as the range is non-negative values, not values less than or equal to 0.
Thus, the correct statement is:
[tex]\[ \text{The domain of the graph is all real numbers less than or equal to 0.} \][/tex]
