Answer :
To determine which statements are true, let's analyze each function in detail.
1. Statement: [tex]\( f(x) = 2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x) = \sqrt{x} \)[/tex].
- Domain Analysis: The domain of [tex]\( f(x) = \sqrt{x} \)[/tex] is [tex]\( x \geq 0 \)[/tex] because we cannot take the square root of a negative number. The function [tex]\( 2 \sqrt{x} \)[/tex] also has the same domain [tex]\( x \geq 0 \)[/tex]. So, the domains are identical.
- Range Analysis: The range of [tex]\( f(x) = \sqrt{x} \)[/tex] is [tex]\( y \geq 0 \)[/tex] because square roots yield non-negative results. For [tex]\( 2 \sqrt{x} \)[/tex], every value of [tex]\( \sqrt{x} \)[/tex] is multiplied by 2, therefore the outputs are also non-negative but scaled up. Thus, the range of [tex]\( 2 \sqrt{x} \)[/tex] is [tex]\( y \geq 0 \)[/tex], but the actual values are different due to the multiplication by 2. So, the ranges are not the same.
- Conclusion: False
2. Statement: [tex]\( f(x) = -2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x) = \sqrt{x} \)[/tex].
- Domain Analysis: Like before, the function [tex]\( -2 \sqrt{x} \)[/tex] has the same domain [tex]\( x \geq 0 \)[/tex].
- Range Analysis: The range of [tex]\( f(x) = \sqrt{x} \)[/tex] is [tex]\( y \geq 0 \)[/tex], but for [tex]\( -2 \sqrt{x} \)[/tex], every value of [tex]\( \sqrt{x} \)[/tex] is multiplied by -2, thereby producing non-positive results. Thus, the range of [tex]\( -2 \sqrt{x} \)[/tex] is [tex]\( y \leq 0 \)[/tex], which is different from [tex]\( f(x) = \sqrt{x} \)[/tex].
- Conclusion: False
3. Statement: [tex]\( f(x) = -\sqrt{x} \)[/tex] has the same domain as [tex]\( f(x) = \sqrt{x} \)[/tex], but a different range.
- Domain Analysis: The domain for both [tex]\( -\sqrt{x} \)[/tex] and [tex]\( \sqrt{x} \)[/tex] is [tex]\( x \geq 0 \)[/tex].
- Range Analysis: The range of [tex]\( \sqrt{x} \)[/tex] is [tex]\( y \geq 0 \)[/tex], while the range of [tex]\( -\sqrt{x} \)[/tex] is [tex]\( y \leq 0 \)[/tex] because the outputs are non-positive (the values of [tex]\( \sqrt{x} \)[/tex] are negated).
- Conclusion: True
4. Statement: [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 Analysis: Both [tex]\( \frac{1}{2} \sqrt{x} \)[/tex] and [tex]\( \sqrt{x} \)[/tex] have the same domain [tex]\( x \geq 0 \)[/tex].
- Range Analysis: The range of [tex]\( \sqrt{x} \)[/tex] is [tex]\( y \geq 0 \)[/tex]. For [tex]\( \frac{1}{2} \sqrt{x} \)[/tex], every value of [tex]\( \sqrt{x} \)[/tex] is multiplied by [tex]\(\frac{1}{2}\)[/tex], producing non-negative results, but scaled down. So, the range is still [tex]\( y \geq 0 \)[/tex], though the actual values are different since they are halved.
- Conclusion: True
Thus, the true statements 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.
Therefore, the indices of the true statements are 3 and 4.
1. Statement: [tex]\( f(x) = 2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x) = \sqrt{x} \)[/tex].
- Domain Analysis: The domain of [tex]\( f(x) = \sqrt{x} \)[/tex] is [tex]\( x \geq 0 \)[/tex] because we cannot take the square root of a negative number. The function [tex]\( 2 \sqrt{x} \)[/tex] also has the same domain [tex]\( x \geq 0 \)[/tex]. So, the domains are identical.
- Range Analysis: The range of [tex]\( f(x) = \sqrt{x} \)[/tex] is [tex]\( y \geq 0 \)[/tex] because square roots yield non-negative results. For [tex]\( 2 \sqrt{x} \)[/tex], every value of [tex]\( \sqrt{x} \)[/tex] is multiplied by 2, therefore the outputs are also non-negative but scaled up. Thus, the range of [tex]\( 2 \sqrt{x} \)[/tex] is [tex]\( y \geq 0 \)[/tex], but the actual values are different due to the multiplication by 2. So, the ranges are not the same.
- Conclusion: False
2. Statement: [tex]\( f(x) = -2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x) = \sqrt{x} \)[/tex].
- Domain Analysis: Like before, the function [tex]\( -2 \sqrt{x} \)[/tex] has the same domain [tex]\( x \geq 0 \)[/tex].
- Range Analysis: The range of [tex]\( f(x) = \sqrt{x} \)[/tex] is [tex]\( y \geq 0 \)[/tex], but for [tex]\( -2 \sqrt{x} \)[/tex], every value of [tex]\( \sqrt{x} \)[/tex] is multiplied by -2, thereby producing non-positive results. Thus, the range of [tex]\( -2 \sqrt{x} \)[/tex] is [tex]\( y \leq 0 \)[/tex], which is different from [tex]\( f(x) = \sqrt{x} \)[/tex].
- Conclusion: False
3. Statement: [tex]\( f(x) = -\sqrt{x} \)[/tex] has the same domain as [tex]\( f(x) = \sqrt{x} \)[/tex], but a different range.
- Domain Analysis: The domain for both [tex]\( -\sqrt{x} \)[/tex] and [tex]\( \sqrt{x} \)[/tex] is [tex]\( x \geq 0 \)[/tex].
- Range Analysis: The range of [tex]\( \sqrt{x} \)[/tex] is [tex]\( y \geq 0 \)[/tex], while the range of [tex]\( -\sqrt{x} \)[/tex] is [tex]\( y \leq 0 \)[/tex] because the outputs are non-positive (the values of [tex]\( \sqrt{x} \)[/tex] are negated).
- Conclusion: True
4. Statement: [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 Analysis: Both [tex]\( \frac{1}{2} \sqrt{x} \)[/tex] and [tex]\( \sqrt{x} \)[/tex] have the same domain [tex]\( x \geq 0 \)[/tex].
- Range Analysis: The range of [tex]\( \sqrt{x} \)[/tex] is [tex]\( y \geq 0 \)[/tex]. For [tex]\( \frac{1}{2} \sqrt{x} \)[/tex], every value of [tex]\( \sqrt{x} \)[/tex] is multiplied by [tex]\(\frac{1}{2}\)[/tex], producing non-negative results, but scaled down. So, the range is still [tex]\( y \geq 0 \)[/tex], though the actual values are different since they are halved.
- Conclusion: True
Thus, the true statements 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.
Therefore, the indices of the true statements are 3 and 4.
