Question

how are the two functions $f(x) = 0.7(6)^x$ and $g(x) = 0.7(6)^{-x}$ related to each other?
$g(x)$ is the reflection of $f(x)$ over the $x$-axis.
$g(x)$ is the reflection of $f(x)$ over both axes.
$g(x)$ is the reflection of $f(x)$ over the $y$-axis.
$g(x)$ and $f(x)$ will appear to be the same function.

Explanation:

Step1: Recall reflection rules

For a function \( y = f(x) \), reflection over \( y \)-axis is \( y = f(-x) \), over \( x \)-axis is \( y=-f(x) \), over both axes is \( y = -f(-x) \).

Step2: Analyze \( g(x) \) and \( f(x) \)

Given \( f(x)=0.7(6)^x \), \( g(x)=0.7(6)^{-x} \). Notice \( g(x)=f(-x) \), which matches reflection over \( y \)-axis rule.

Answer:

\( g(x) \) is the reflection of \( f(x) \) over the \( y \)-axis.