Which function represents a reflection of [tex]$f(x)=\frac{3}{8}(4)^x$[/tex] across the [tex]y[/tex]-axis?

A. [tex]g(x)=-\frac{3}{8}\left(\frac{1}{4}\right)^x[/tex]
B. [tex]g(x)=-\frac{3}{8}(4)^x[/tex]
C. [tex]g(x)=\frac{8}{3}(4)^{-x}[/tex]
D. [tex]g(x)=\frac{3}{8}(4)^{-x}[/tex]



Answer :

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.