To determine the function that represents a reflection of \(f(x) = \frac{3}{8}(4^x)\) across the \(y\)-axis, we need to understand what this transformation entails. Reflecting a function \(f(x)\) across the \(y\)-axis involves replacing \(x\) with \(-x\).
Step-by-step:
1. Original Function:
[tex]\[ f(x) = \frac{3}{8}(4^x) \][/tex]
2. Reflection Across the \(y\)-Axis:
To reflect this function across the \(y\)-axis, we replace \(x\) with \(-x\):
[tex]\[ f(-x) = \frac{3}{8}(4^{-x}) \][/tex]
3. Simplifying the Expression:
The expression \(4^{-x}\) can be directly incorporated since it represents the negative exponent transformation.
So, the function representing the reflection is:
[tex]\[ g(x) = \frac{3}{8}(4^{-x}) \][/tex]
Comparing this result with the given options:
1. \( g(x) = -\frac{3}{8}\left(\frac{1}{4}\right)^x \)
2. \( g(x) = -\frac{3}{8}(4)^x \)
3. \( g(x) = \frac{8}{3}(4)^{-x} \)
4. \( g(x) = \frac{3}{8}(4)^{-x} \)
The correct match is:
[tex]\[ g(x) = \frac{3}{8}(4)^{-x} \][/tex]
Therefore, the function that represents the reflection of [tex]\(f(x)\)[/tex] across the [tex]\(y\)[/tex]-axis is given by option 4.
