To analyze the function [tex]\( f(x) = -\sqrt{-x} \)[/tex] and determine the correctness of the given statements, let's consider the following points in a step-by-step manner:
### Step 1: Determine the Domain of [tex]\( f(x) \)[/tex]
The domain of a function is the set of all possible input values (x-values) that the function can accept without resulting in any undefined or non-real values. For the function [tex]\( f(x) = -\sqrt{-x} \)[/tex], the expression inside the square root must be non-negative.
Inside the square root, we have [tex]\(-x\)[/tex]. Therefore, for the square root to be defined:
[tex]\[ -x \geq 0 \][/tex]
Solving the above inequality, we get:
[tex]\[ x \leq 0 \][/tex]
This means the domain of the function [tex]\( f(x) = -\sqrt{-x} \)[/tex] is all real numbers less than or equal to zero. In other words:
[tex]\[ \text{Domain} = (-\infty, 0] \][/tex]
### Step 2: Determine the Range of [tex]\( f(x) \)[/tex]
The range of a function is the set of all possible output values (y-values) that the function can produce. Let's consider the function [tex]\( f(x) = -\sqrt{-x} \)[/tex] more closely:
- The square root function [tex]\(\sqrt{-x}\)[/tex] for [tex]\( x \leq 0 \)[/tex] always results in a non-negative value (greater than or equal to zero).
However, because of the negative sign in front of the square root, the output of the function [tex]\( f(x) = -\sqrt{-x} \)[/tex] will always be non-positive. Thus:
[tex]\[ f(x) \leq 0 \][/tex]
Therefore, the range of [tex]\( f(x) \)[/tex] includes all values that are less than or equal to 0 (i.e., non-positive values):
[tex]\[ \text{Range} = (-\infty, 0] \][/tex]
### Step 3: Analyze Each Statement
Let’s evaluate the provided statements in light of the domain and the range determined above:
1. The domain of the function is all negative real numbers.
This statement is false because the domain is [tex]\( (-\infty, 0] \)[/tex], which includes zero in addition to all negative real numbers.
2. The range of the function is all positive real numbers.
This statement is false because the range is [tex]\( (-\infty, 0] \)[/tex], which includes zero and negative real numbers, not positive real numbers.
3. The domain and range of the function have opposite signs.
This statement is true. The domain is [tex]\( (-\infty, 0] \)[/tex], which consists of zero and negative numbers, while the range is also [tex]\( (-\infty, 0] \)[/tex], which consists of zero and negative numbers, so their signs effectively match as non-positive.
4. The domain and range of the function are the same.
This statement is true. Both the domain and range are [tex]\( (-\infty, 0] \)[/tex].
### Conclusion
Based on the above analysis, the correct statements are:
- The domain and range of the function have opposite signs.
- The domain and range of the function are the same.
These statements properly describe the characteristics of the function [tex]\( f(x) = -\sqrt{-x} \)[/tex].
