To determine which statement is true about the function [tex]\( f(x) = -\sqrt{x} \)[/tex], let's analyze and compare the domains and ranges of the three functions given: [tex]\( f(x) = -\sqrt{x} \)[/tex], [tex]\( f(x) = \sqrt{x} \)[/tex], and [tex]\( f(x) = -\sqrt{-x} \)[/tex].
### 1. Function [tex]\( f(x) = -\sqrt{x} \)[/tex]
- Domain: [tex]\(x \geq 0\)[/tex]
- The square root function is defined for non-negative values of [tex]\(x\)[/tex]. Hence, [tex]\( x \geq 0 \)[/tex].
- Range: [tex]\( y \leq 0 \)[/tex]
- The square root function yields non-negative values, and since we are negating it, the range will be non-positive values.
### 2. Function [tex]\( f(x) = \sqrt{x} \)[/tex]
- Domain: [tex]\(x \geq 0\)[/tex]
- This function is defined for non-negative values of [tex]\(x\)[/tex]. Hence, [tex]\( x \geq 0 \)[/tex].
- Range: [tex]\( y \geq 0 \)[/tex]
- The square root function yields non-negative values.
### 3. Function [tex]\( f(x) = -\sqrt{-x} \)[/tex]
- Domain: [tex]\(x \leq 0\)[/tex]
- The square root function is defined for non-negative values of the input. Since the input is [tex]\(-x\)[/tex], it means [tex]\( -x \geq 0 \)[/tex] or [tex]\( x \leq 0 \)[/tex].
- Range: [tex]\( y \leq 0 \)[/tex]
- Negating the square root function means the output will be non-positive.
### Comparisons
#### Comparing [tex]\( f(x) = -\sqrt{x} \)[/tex] and [tex]\( f(x) = \sqrt{x} \)[/tex]
- Domain: Both functions have the same domain [tex]\( x \geq 0 \)[/tex].
- Range: The ranges are different. [tex]\( f(x) = -\sqrt{x} \)[/tex] has a range [tex]\( y \leq 0 \)[/tex], whereas [tex]\( f(x) = \sqrt{x} \)[/tex] has a range [tex]\( y \geq 0 \)[/tex].
Since they do not share the same range, the first statement (same domain and range) is false. The second statement (same range but not the same domain) is also false.
#### Comparing [tex]\( f(x) = -\sqrt{x} \)[/tex] and [tex]\( f(x) = -\sqrt{-x} \)[/tex]
- Domain: The domains are different. [tex]\( f(x) = -\sqrt{x} \)[/tex] has a domain [tex]\( x \geq 0 \)[/tex], whereas [tex]\( f(x) = -\sqrt{-x} \)[/tex] has a domain [tex]\( x \leq 0 \)[/tex].
- Range: Both functions have the same range [tex]\( y \leq 0 \)[/tex].
Since they do not share the same domain but share the same range, the third statement (same domain and range) is false. The fourth statement (same range but not the same domain) is true.
Thus, the correct statement is:
- It has the same range but not the same domain as the function [tex]\( f(x) = -\sqrt{-x} \)[/tex].
