At Westonci.ca, we make it easy to get the answers you need from a community of informed and experienced contributors. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
To solve this problem, we'll need to understand the reflection of the function [tex]\( f(x) = \sqrt{x} \)[/tex] over the [tex]\( y \)[/tex]-axis. Let's start from the transformation basics and break down the process to identify which set of points belongs to the reflected function correctly.
### Understanding the Reflection
The original function is given by:
[tex]\[ f(x) = \sqrt{x} \][/tex]
When this graph is reflected over the [tex]\( y \)[/tex]-axis, the function transforms in the following way:
[tex]\[ f(-x) = \sqrt{-x} \][/tex]
### Identifying Valid Points
For the reflected function [tex]\( f(-x) = \sqrt{-x} \)[/tex] to be real-valued and defined, [tex]\( -x \)[/tex] should be non-negative, meaning [tex]\( x \)[/tex] must be non-positive (i.e., [tex]\( x \leq 0 \)[/tex]). Thus, all valid [tex]\( x \)[/tex] values for which [tex]\( \sqrt{-x} \)[/tex] is real must be negative.
Let's check if the provided points satisfy this condition. We have four choices:
#### Choice 1: [tex]\((-81, 9), (-36, 6), (-1, 1)\)[/tex]
- For [tex]\( \sqrt{-(-81)} = 9 \Rightarrow \sqrt{81} = 9 \)[/tex] (Valid)
- For [tex]\( \sqrt{-(-36)} = 6 \Rightarrow \sqrt{36} = 6 \)[/tex] (Valid)
- For [tex]\( \sqrt{-(-1)} = 1 \Rightarrow \sqrt{1} = 1 \)[/tex] (Valid)
All three points satisfy the condition and lie on the reflection [tex]\( f(-x) = \sqrt{-x} \)[/tex].
#### Choice 2: [tex]\((4, -1), (16, -4), (36, -6)\)[/tex]
- For [tex]\( \sqrt{-(4)} \)[/tex], [tex]\( -4 \)[/tex] is not a valid input for [tex]\( \sqrt{} \)[/tex] function because it's a complex number.
- Similar issues arise for points [tex]\( (16, -4) \)[/tex] and [tex]\( (36, -6) \)[/tex].
These points do not satisfy the condition.
#### Choice 3: [tex]\((-49, 7), (-18, 9), (-1, 1)\)[/tex]
- For [tex]\( \sqrt{-(-49)} = 7 \Rightarrow \sqrt{49} = 7 \)[/tex] (Valid)
- For [tex]\( \sqrt{-(-18)} = 9 \Rightarrow \sqrt{18} \neq 9 \)[/tex] (Invalid as [tex]\(\sqrt{18}\approx4.24\)[/tex])
- Checking the remaining doesn't matter as one invalid point already disqualifies this set.
These points are not all valid.
#### Choice 4: [tex]\((1, -1), (4, -16), (5, -25)\)[/tex]
- For [tex]\( \sqrt{-(1)} \)[/tex], [tex]\( -1 \)[/tex] is again not a valid input similarly.
- Same problem exists for points [tex]\( (4, -16) \)[/tex] and [tex]\( (5, -25) \)[/tex].
These points are incorrect as well.
### Conclusion
The set of points that correctly lies on the graph of the reflection [tex]\( f(-x) = \sqrt{-x} \)[/tex] is:
[tex]\[ \boxed{(-81, 9), (-36, 6), (-1, 1)} \][/tex]
Therefore, the set of points in Choice 1 is the correct answer.
### Understanding the Reflection
The original function is given by:
[tex]\[ f(x) = \sqrt{x} \][/tex]
When this graph is reflected over the [tex]\( y \)[/tex]-axis, the function transforms in the following way:
[tex]\[ f(-x) = \sqrt{-x} \][/tex]
### Identifying Valid Points
For the reflected function [tex]\( f(-x) = \sqrt{-x} \)[/tex] to be real-valued and defined, [tex]\( -x \)[/tex] should be non-negative, meaning [tex]\( x \)[/tex] must be non-positive (i.e., [tex]\( x \leq 0 \)[/tex]). Thus, all valid [tex]\( x \)[/tex] values for which [tex]\( \sqrt{-x} \)[/tex] is real must be negative.
Let's check if the provided points satisfy this condition. We have four choices:
#### Choice 1: [tex]\((-81, 9), (-36, 6), (-1, 1)\)[/tex]
- For [tex]\( \sqrt{-(-81)} = 9 \Rightarrow \sqrt{81} = 9 \)[/tex] (Valid)
- For [tex]\( \sqrt{-(-36)} = 6 \Rightarrow \sqrt{36} = 6 \)[/tex] (Valid)
- For [tex]\( \sqrt{-(-1)} = 1 \Rightarrow \sqrt{1} = 1 \)[/tex] (Valid)
All three points satisfy the condition and lie on the reflection [tex]\( f(-x) = \sqrt{-x} \)[/tex].
#### Choice 2: [tex]\((4, -1), (16, -4), (36, -6)\)[/tex]
- For [tex]\( \sqrt{-(4)} \)[/tex], [tex]\( -4 \)[/tex] is not a valid input for [tex]\( \sqrt{} \)[/tex] function because it's a complex number.
- Similar issues arise for points [tex]\( (16, -4) \)[/tex] and [tex]\( (36, -6) \)[/tex].
These points do not satisfy the condition.
#### Choice 3: [tex]\((-49, 7), (-18, 9), (-1, 1)\)[/tex]
- For [tex]\( \sqrt{-(-49)} = 7 \Rightarrow \sqrt{49} = 7 \)[/tex] (Valid)
- For [tex]\( \sqrt{-(-18)} = 9 \Rightarrow \sqrt{18} \neq 9 \)[/tex] (Invalid as [tex]\(\sqrt{18}\approx4.24\)[/tex])
- Checking the remaining doesn't matter as one invalid point already disqualifies this set.
These points are not all valid.
#### Choice 4: [tex]\((1, -1), (4, -16), (5, -25)\)[/tex]
- For [tex]\( \sqrt{-(1)} \)[/tex], [tex]\( -1 \)[/tex] is again not a valid input similarly.
- Same problem exists for points [tex]\( (4, -16) \)[/tex] and [tex]\( (5, -25) \)[/tex].
These points are incorrect as well.
### Conclusion
The set of points that correctly lies on the graph of the reflection [tex]\( f(-x) = \sqrt{-x} \)[/tex] is:
[tex]\[ \boxed{(-81, 9), (-36, 6), (-1, 1)} \][/tex]
Therefore, the set of points in Choice 1 is the correct answer.
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
