To determine which statements are true, we must analyze the domain and range of each given function and compare each to [tex]\( f(x) = \sqrt{x} \)[/tex].
1. [tex]\( f(x) = 2 \sqrt{x} \)[/tex] vs [tex]\( f(x) = \sqrt{x} \)[/tex]
- Domain: Both functions are defined for [tex]\( x \geq 0 \)[/tex] because the square root function is only defined for non-negative values.
Domain: [tex]\( x \geq 0 \)[/tex]
- Range: For [tex]\( f(x) = \sqrt{x} \)[/tex], the range is [tex]\( y \geq 0 \)[/tex]. For [tex]\( f(x) = 2 \sqrt{x} \)[/tex], the output values are double those of [tex]\( \sqrt{x} \)[/tex]. Therefore, the range is still [tex]\( y \geq 0 \)[/tex], but it spans different numeric values.
Range: [tex]\( f(x) = \sqrt{x} \rightarrow [0, \infty) \)[/tex]
[tex]\( f(x) = 2 \sqrt{x} \rightarrow [0, \infty) \)[/tex]
Since the domain is the same but the numeric output values (range) differ despite spanning the same interval, this statement is false.
2. [tex]\( f(x) = -2 \sqrt{x} \)[/tex] vs [tex]\( f(x) = \sqrt{x} \)[/tex]
- Domain: Both functions are defined for [tex]\( x \geq 0 \)[/tex].
Domain: [tex]\( x \geq 0 \)[/tex]
- Range: For [tex]\( f(x) = \sqrt{x} \)[/tex], the range is [tex]\( y \geq 0 \)[/tex]. For [tex]\( f(x) = -2 \sqrt{x} \)[/tex], the output is negative, therefore the range is [tex]\( y \leq 0 \)[/tex].
Range: [tex]\( f(x) = \sqrt{x} \rightarrow [0, \infty) \)[/tex]
[tex]\( f(x) = -2 \sqrt{x} \rightarrow (-\infty, 0] \)[/tex]
Since both the domain is the same but the range is entirely different, this statement is false.
3. [tex]\( f(x) = -\sqrt{x} \)[/tex] vs [tex]\( f(x) = \sqrt{x} \)[/tex]
- Domain: Both functions are defined for [tex]\( x \geq 0 \)[/tex].
Domain: [tex]\( x \geq 0 \)[/tex]
- Range: For [tex]\( f(x) = \sqrt{x} \)[/tex], the range is [tex]\( y \geq 0 \)[/tex]. For [tex]\( f(x) = -\sqrt{x} \)[/tex], the output is negative, therefore the range is [tex]\( y \leq 0 \)[/tex].
Range: [tex]\( f(x) = \sqrt{x} \rightarrow [0, \infty) \)[/tex]
[tex]\( f(x) = -\sqrt{x} \rightarrow (-\infty, 0] \)[/tex]
Since the domain is the same but the range is entirely different, this statement is true.
4. [tex]\( f(x) = \frac{1}{2} \sqrt{x} \)[/tex] vs [tex]\( f(x) = \sqrt{x} \)[/tex]
- Domain: Both functions are defined for [tex]\( x \geq 0 \)[/tex].
Domain: [tex]\( x \geq 0 \)[/tex]
- Range: For [tex]\( f(x) = \sqrt{x} \)[/tex], the range is [tex]\( y \geq 0 \)[/tex]. For [tex]\( f(x) = \frac{1}{2} \sqrt{x} \)[/tex], the output values are scaled down by a factor of [tex]\( \frac{1}{2} \)[/tex], so the range is still [tex]\( y \geq 0 \)[/tex], but with different numeric values.
Range: [tex]\( f(x) = \sqrt{x} \rightarrow [0, \infty) \)[/tex]
[tex]\( f(x) = \frac{1}{2} \sqrt{x} \rightarrow [0, \infty) \)[/tex]
Since the domain is the same but the numeric output values (range) differ despite spanning the same interval, this statement is true.
Therefore, the statements that are true are:
- [tex]\( f(x) = -\sqrt{x} \)[/tex] has the same domain as [tex]\( f(x) = \sqrt{x} \)[/tex], but a different range.
- [tex]\( f(x) = \frac{1}{2} \sqrt{x} \)[/tex] has the same domain as [tex]\( f(x) = \sqrt{x} \)[/tex], but a different range.
So, the true statements are 3 and 4.
