Westonci.ca is your trusted source for accurate answers to all your questions. Join our community and start learning today! Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
Let's analyze each function and compare their domains and ranges with that of [tex]\( f(x) = \sqrt{x} \)[/tex].
1. [tex]\( f(x) = 2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x) = \sqrt{x} \)[/tex].
Domain:
- [tex]\( f(x) = \sqrt{x} \)[/tex] is defined for [tex]\( x \geq 0 \)[/tex].
- [tex]\( f(x) = 2\sqrt{x} \)[/tex] is also defined for [tex]\( x \geq 0 \)[/tex].
Range:
- [tex]\( f(x) = \sqrt{x} \)[/tex] has a range of [tex]\([0, \infty)\)[/tex].
- [tex]\( f(x) = 2\sqrt{x} \)[/tex] has a range of [tex]\([0, \infty)\)[/tex] because multiplying by 2 doesn't change the non-negative nature of the output values.
Therefore, this statement is true.
2. [tex]\( f(x) = -2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x) = \sqrt{x} \)[/tex].
Domain:
- [tex]\( f(x) = \sqrt{x} \)[/tex] is defined for [tex]\( x \geq 0 \)[/tex].
- [tex]\( f(x) = -2\sqrt{x} \)[/tex] is also defined for [tex]\( x \geq 0 \)[/tex].
Range:
- [tex]\( f(x) = \sqrt{x} \)[/tex] has a range of [tex]\([0, \infty)\)[/tex].
- [tex]\( f(x) = -2\sqrt{x} \)[/tex] has a range of [tex]\((-\infty, 0] \)[/tex] because multiplying by -2 flips the range to cover all non-positive values.
Therefore, this statement is false.
3. [tex]\( f(x) = -\sqrt{x} \)[/tex] has the same domain as [tex]\( f(x) = \sqrt{x} \)[/tex], but a different range.
Domain:
- [tex]\( f(x) = \sqrt{x} \)[/tex] is defined for [tex]\( x \geq 0 \)[/tex].
- [tex]\( f(x) = -\sqrt{x} \)[/tex] is also defined for [tex]\( x \geq 0 \)[/tex].
Range:
- [tex]\( f(x) = \sqrt{x} \)[/tex] has a range of [tex]\([0, \infty)\)[/tex].
- [tex]\( f(x) = -\sqrt{x} \)[/tex] has a range of [tex]\((-\infty, 0] \)[/tex].
Therefore, this statement is true.
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:
- [tex]\( f(x) = \sqrt{x} \)[/tex] is defined for [tex]\( x \geq 0 \)[/tex].
- [tex]\( f(x) = \frac{1}{2} \sqrt{x} \)[/tex] is also defined for [tex]\( x \geq 0 \)[/tex].
Range:
- [tex]\( f(x) = \sqrt{x} \)[/tex] has a range of [tex]\([0, \infty)\)[/tex].
- [tex]\( f(x) = \frac{1}{2}\sqrt{x} \)[/tex] has a range of [tex]\([0, \infty)\)[/tex], but the values are scaled down by a factor of [tex]\(\frac{1}{2}\)[/tex].
Therefore, this statement is true.
Final Answer:
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.
So, the correct checkboxes are:
- ✓ [tex]\( f(x)=2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x)=\sqrt{x} \)[/tex].
- [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.
1. [tex]\( f(x) = 2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x) = \sqrt{x} \)[/tex].
Domain:
- [tex]\( f(x) = \sqrt{x} \)[/tex] is defined for [tex]\( x \geq 0 \)[/tex].
- [tex]\( f(x) = 2\sqrt{x} \)[/tex] is also defined for [tex]\( x \geq 0 \)[/tex].
Range:
- [tex]\( f(x) = \sqrt{x} \)[/tex] has a range of [tex]\([0, \infty)\)[/tex].
- [tex]\( f(x) = 2\sqrt{x} \)[/tex] has a range of [tex]\([0, \infty)\)[/tex] because multiplying by 2 doesn't change the non-negative nature of the output values.
Therefore, this statement is true.
2. [tex]\( f(x) = -2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x) = \sqrt{x} \)[/tex].
Domain:
- [tex]\( f(x) = \sqrt{x} \)[/tex] is defined for [tex]\( x \geq 0 \)[/tex].
- [tex]\( f(x) = -2\sqrt{x} \)[/tex] is also defined for [tex]\( x \geq 0 \)[/tex].
Range:
- [tex]\( f(x) = \sqrt{x} \)[/tex] has a range of [tex]\([0, \infty)\)[/tex].
- [tex]\( f(x) = -2\sqrt{x} \)[/tex] has a range of [tex]\((-\infty, 0] \)[/tex] because multiplying by -2 flips the range to cover all non-positive values.
Therefore, this statement is false.
3. [tex]\( f(x) = -\sqrt{x} \)[/tex] has the same domain as [tex]\( f(x) = \sqrt{x} \)[/tex], but a different range.
Domain:
- [tex]\( f(x) = \sqrt{x} \)[/tex] is defined for [tex]\( x \geq 0 \)[/tex].
- [tex]\( f(x) = -\sqrt{x} \)[/tex] is also defined for [tex]\( x \geq 0 \)[/tex].
Range:
- [tex]\( f(x) = \sqrt{x} \)[/tex] has a range of [tex]\([0, \infty)\)[/tex].
- [tex]\( f(x) = -\sqrt{x} \)[/tex] has a range of [tex]\((-\infty, 0] \)[/tex].
Therefore, this statement is true.
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:
- [tex]\( f(x) = \sqrt{x} \)[/tex] is defined for [tex]\( x \geq 0 \)[/tex].
- [tex]\( f(x) = \frac{1}{2} \sqrt{x} \)[/tex] is also defined for [tex]\( x \geq 0 \)[/tex].
Range:
- [tex]\( f(x) = \sqrt{x} \)[/tex] has a range of [tex]\([0, \infty)\)[/tex].
- [tex]\( f(x) = \frac{1}{2}\sqrt{x} \)[/tex] has a range of [tex]\([0, \infty)\)[/tex], but the values are scaled down by a factor of [tex]\(\frac{1}{2}\)[/tex].
Therefore, this statement is true.
Final Answer:
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.
So, the correct checkboxes are:
- ✓ [tex]\( f(x)=2 \sqrt{x} \)[/tex] has the same domain and range as [tex]\( f(x)=\sqrt{x} \)[/tex].
- [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.
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
