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.
