Discover answers to your most pressing questions at Westonci.ca, the ultimate Q&A platform that connects you with expert solutions. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
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.
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.
