To determine which function represents the reflection of [tex]\( f(x) = \frac{3}{8} (4)^x \)[/tex] across the [tex]\( y \)[/tex]-axis, we need to understand the concept of reflection and how it affects the function's equation.
### Step-by-Step Solution:
1. Understand Reflection across the [tex]\( y \)[/tex]-Axis:
Reflecting a function [tex]\( f(x) \)[/tex] across the [tex]\( y \)[/tex]-axis involves replacing [tex]\( x \)[/tex] with [tex]\( -x \)[/tex] in the function's equation. Thus, for the function [tex]\( f(x) = \frac{3}{8} (4)^x \)[/tex], we should replace [tex]\( x \)[/tex] with [tex]\( -x \)[/tex].
2. Apply the Transformation:
Let's apply the transformation [tex]\( x \rightarrow -x \)[/tex] to the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(-x) = \frac{3}{8} (4)^{-x}
\][/tex]
3. Simplify the Expression (if needed, but in this case, we're already in a simplified form):
The expression [tex]\( \frac{3}{8} (4)^{-x} \)[/tex] is already simplified. We don't need to perform additional simplifications.
4. Match with Given Options:
Now, we need to compare the transformed function [tex]\( f(-x) = \frac{3}{8} (4)^{-x} \)[/tex] with the provided options:
- [tex]\( g(x) = -\frac{3}{8} \left(\frac{1}{4}\right)^x \)[/tex]
- [tex]\( g(x) = -\frac{3}{8} (4)^x \)[/tex]
- [tex]\( g(x) = \frac{8}{3} (4)^{-x} \)[/tex]
- [tex]\( g(x) = \frac{3}{8} (4)^{-x} \)[/tex]
By comparing, we see that the function [tex]\( g(x) = \frac{3}{8} (4)^{-x} \)[/tex] (option 4) matches exactly with our transformed function.
5. Identify the Corresponding Option:
The function that represents the reflection of [tex]\( f(x) \)[/tex] across the [tex]\( y \)[/tex]-axis is [tex]\( g(x) = \frac{3}{8} (4)^{-x} \)[/tex], corresponding to option 4.
Thus, the function that represents a reflection of [tex]\( f(x) = \frac{3}{8} (4)^x \)[/tex] across the [tex]\( y \)[/tex]-axis is:
[tex]\[ \boxed{g(x) = \frac{3}{8} (4)^{-x}} \][/tex]
### Conclusion:
The correct option is option 4, and the numerical value of the reflected function evaluated at [tex]\( x = 1 \)[/tex] is [tex]\( 0.09375 \)[/tex], confirming that the matching function in the options is [tex]\( g(x) = \frac{3}{8} (4)^{-x} \)[/tex].
