Answer :
Alright, let's examine each statement and determine whether it's true, based on our understanding of the function transformations involving square roots.
1. Statement 1: [tex]\( f(x) = 2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x) = \sqrt{x} \)[/tex].
- Domain: The domain of [tex]\( \sqrt{x} \)[/tex] is all [tex]\( x \geq 0 \)[/tex], as negative values under the square root are not defined in the real number system. Thus, the domain of [tex]\( 2 \sqrt{x} \)[/tex] is also [tex]\( x \geq 0 \)[/tex]. So, the domains are the same.
- Range: The range of [tex]\( \sqrt{x} \)[/tex] is [tex]\([0, \infty)\)[/tex] because the square root function produces non-negative values. The range of [tex]\( 2 \sqrt{x} \)[/tex] is also [tex]\([0, \infty)\)[/tex] because multiplying by 2 still produces non-negative values, but they are scaled by a factor of 2.
Thus, the statement that [tex]\( f(x) = 2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x) = \sqrt{x} \)[/tex] is true.
2. Statement 2: [tex]\( f(x) = -2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x) = \sqrt{x} \)[/tex].
- Domain: As before, the domain of [tex]\( \sqrt{x} \)[/tex] and hence [tex]\( -2 \sqrt{x} \)[/tex] is [tex]\( x \geq 0 \)[/tex], so the domains are the same.
- Range: The range of [tex]\( \sqrt{x} \)[/tex] is [tex]\([0, \infty)\)[/tex]. However, the range of [tex]\( -2 \sqrt{x} \)[/tex] will be all negative values, [tex]\( (-\infty, 0] \)[/tex], because multiplying by -2 takes the non-negative outputs of [tex]\( \sqrt{x} \)[/tex] and makes them non-positive.
Thus, the statement that [tex]\( f(x) = -2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x) = \sqrt{x} \)[/tex] is false.
3. Statement 3: [tex]\( f(x) = -\sqrt{x} \)[/tex] has the same domain as [tex]\( f(x) = \sqrt{x} \)[/tex], but a different range.
- Domain: Both functions [tex]\( \sqrt{x} \)[/tex] and [tex]\( -\sqrt{x} \)[/tex] have the domain [tex]\( x \geq 0 \)[/tex].
- Range: The range of [tex]\( \sqrt{x} \)[/tex] is [tex]\([0, \infty)\)[/tex]. The range of [tex]\( -\sqrt{x} \)[/tex] is [tex]\( (-\infty, 0] \)[/tex], as it makes all outputs non-positive.
Therefore, the statement that [tex]\( f(x) = -\sqrt{x} \)[/tex] has the same domain as [tex]\( f(x) = \sqrt{x} \)[/tex], but a different range is true.
4. Statement 4: [tex]\( f(x) = \frac{1}{2} \sqrt{x} \)[/tex] has the same domain as [tex]\( f(x) = \sqrt{x} \)[/tex], but a different range.
- Domain: Both functions [tex]\( \sqrt{x} \)[/tex] and [tex]\( \frac{1}{2} \sqrt{x} \)[/tex] have the domain [tex]\( x \geq 0 \)[/tex].
- Range: The range of [tex]\( \sqrt{x} \)[/tex] is [tex]\([0, \infty)\)[/tex]. The range of [tex]\( \frac{1}{2} \sqrt{x} \)[/tex] is also [tex]\([0, \infty)\)[/tex] but scaled by a factor of [tex]\(\frac{1}{2}\)[/tex]. However, the essential non-negative nature of the output remains, making the effective ranges the same in nature but potentially different in specific values.
Thus, the statement that [tex]\( f(x) = \frac{1}{2} \sqrt{x} \)[/tex] has the same domain as [tex]\( f(x) = \sqrt{x} \)[/tex], but a different range is true.
In conclusion, the statements that are true are:
- [tex]\( f(x) = 2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x) = \sqrt{x} \)[/tex].
- [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.
These correspond to statements 1, 3, and 4.
1. Statement 1: [tex]\( f(x) = 2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x) = \sqrt{x} \)[/tex].
- Domain: The domain of [tex]\( \sqrt{x} \)[/tex] is all [tex]\( x \geq 0 \)[/tex], as negative values under the square root are not defined in the real number system. Thus, the domain of [tex]\( 2 \sqrt{x} \)[/tex] is also [tex]\( x \geq 0 \)[/tex]. So, the domains are the same.
- Range: The range of [tex]\( \sqrt{x} \)[/tex] is [tex]\([0, \infty)\)[/tex] because the square root function produces non-negative values. The range of [tex]\( 2 \sqrt{x} \)[/tex] is also [tex]\([0, \infty)\)[/tex] because multiplying by 2 still produces non-negative values, but they are scaled by a factor of 2.
Thus, the statement that [tex]\( f(x) = 2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x) = \sqrt{x} \)[/tex] is true.
2. Statement 2: [tex]\( f(x) = -2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x) = \sqrt{x} \)[/tex].
- Domain: As before, the domain of [tex]\( \sqrt{x} \)[/tex] and hence [tex]\( -2 \sqrt{x} \)[/tex] is [tex]\( x \geq 0 \)[/tex], so the domains are the same.
- Range: The range of [tex]\( \sqrt{x} \)[/tex] is [tex]\([0, \infty)\)[/tex]. However, the range of [tex]\( -2 \sqrt{x} \)[/tex] will be all negative values, [tex]\( (-\infty, 0] \)[/tex], because multiplying by -2 takes the non-negative outputs of [tex]\( \sqrt{x} \)[/tex] and makes them non-positive.
Thus, the statement that [tex]\( f(x) = -2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x) = \sqrt{x} \)[/tex] is false.
3. Statement 3: [tex]\( f(x) = -\sqrt{x} \)[/tex] has the same domain as [tex]\( f(x) = \sqrt{x} \)[/tex], but a different range.
- Domain: Both functions [tex]\( \sqrt{x} \)[/tex] and [tex]\( -\sqrt{x} \)[/tex] have the domain [tex]\( x \geq 0 \)[/tex].
- Range: The range of [tex]\( \sqrt{x} \)[/tex] is [tex]\([0, \infty)\)[/tex]. The range of [tex]\( -\sqrt{x} \)[/tex] is [tex]\( (-\infty, 0] \)[/tex], as it makes all outputs non-positive.
Therefore, the statement that [tex]\( f(x) = -\sqrt{x} \)[/tex] has the same domain as [tex]\( f(x) = \sqrt{x} \)[/tex], but a different range is true.
4. Statement 4: [tex]\( f(x) = \frac{1}{2} \sqrt{x} \)[/tex] has the same domain as [tex]\( f(x) = \sqrt{x} \)[/tex], but a different range.
- Domain: Both functions [tex]\( \sqrt{x} \)[/tex] and [tex]\( \frac{1}{2} \sqrt{x} \)[/tex] have the domain [tex]\( x \geq 0 \)[/tex].
- Range: The range of [tex]\( \sqrt{x} \)[/tex] is [tex]\([0, \infty)\)[/tex]. The range of [tex]\( \frac{1}{2} \sqrt{x} \)[/tex] is also [tex]\([0, \infty)\)[/tex] but scaled by a factor of [tex]\(\frac{1}{2}\)[/tex]. However, the essential non-negative nature of the output remains, making the effective ranges the same in nature but potentially different in specific values.
Thus, the statement that [tex]\( f(x) = \frac{1}{2} \sqrt{x} \)[/tex] has the same domain as [tex]\( f(x) = \sqrt{x} \)[/tex], but a different range is true.
In conclusion, the statements that are true are:
- [tex]\( f(x) = 2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x) = \sqrt{x} \)[/tex].
- [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.
These correspond to statements 1, 3, and 4.
