Let's first understand the given functions and then evaluate [tex]\( g(x) \)[/tex] at the specified points.
### Given Functions
1. Original function [tex]\( f(x) \)[/tex]
[tex]\[
f(x) = \frac{1}{6} \left( \frac{2}{5} \right)^x
\][/tex]
2. Reflected function [tex]\( g(x) \)[/tex] across the [tex]\( y \)[/tex]-axis
[tex]\[
g(x) = f(-x) = \frac{1}{6} \left( \frac{2}{5} \right)^{-x}
\][/tex]
### Evaluations at Specified Points
Now, we will calculate [tex]\( g(x) \)[/tex] for the points [tex]\((-3, \frac{4}{375}), (-2, \frac{25}{24}), (2, \frac{2}{75}), (3, -\frac{125}{48})\)[/tex].
#### Point 1: [tex]\( x = -3 \)[/tex]
[tex]\[
g(-3) = f(3) = \frac{1}{6} \left( \frac{2}{5} \right)^3 = \frac{1}{6} \left( \frac{8}{125} \right) = \frac{8}{750} = \frac{4}{375}
\][/tex]
So, [tex]\( g(-3) = \frac{4}{375} \)[/tex].
Comparing [tex]\( (-3, \frac{4}{375}) \)[/tex]:
[tex]\[
\left(-3, \frac{4}{375}\right)
\][/tex]
#### Point 2: [tex]\( x = -2 \)[/tex]
[tex]\[
g(-2) = f(2) = \frac{1}{6} \left( \frac{2}{5} \right)^2 = \frac{1}{6} \left( \frac{4}{25} \right) = \frac{4}{150} = \frac{2}{75}
\][/tex]
So, [tex]\( g(-2) = \frac{2}{75} \)[/tex].
However, the provided [tex]\( y \)[/tex]-value for [tex]\((-2, \frac{25}{24})\)[/tex] doesn’t match this result.
#### Point 3: [tex]\( x = 2 \)[/tex]
[tex]\[
g(2) = f(-2) = \frac{1}{6} \left( \frac{2}{5} \right)^{-2} = \frac{1}{6} \left( \frac{25}{4} \right) = \frac{25}{24}
\][/tex]
So, [tex]\( g(2) = \frac{25}{24} \)[/tex].
Comparing [tex]\( (2, \frac{25}{24}) \)[/tex]:
[tex]\[
\left(2, \frac{25}{24}\right)
\][/tex]
#### Point 4: [tex]\( x = 3 \)[/tex]
[tex]\[
g(3) = f(-3) = \frac{1}{6} \left( \frac{2}{5} \right)^{-3} = \frac{1}{6} \left( \frac{125}{8} \right) = \frac{125}{48}
\][/tex]
So, [tex]\( g(3) = \frac{125}{48} \)[/tex].
However, the provided [tex]\( y \)[/tex]-value for [tex]\( (3, -\frac{125}{48}) \)[/tex] must be corrected.
### Summary:
1. [tex]\((-3, \frac{4}{375})\)[/tex] is correct.
2. [tex]\((-2, \frac{25}{24})\)[/tex] does not match the evaluated result.
3. [tex]\((2, \frac{25}{24})\)[/tex] does match the result at [tex]\( x = 2 \)[/tex].
4. [tex]\((3, -\frac{125}{48})\)[/tex] does not match the evaluated result.
Thus, the confirmed matching points are [tex]\( (-3, \frac{4}{375}) \)[/tex] and [tex]\( (2, \frac{25}{24}) \)[/tex].
