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.
$g(x)$ and $f(x)$ will appear to be the same function.
$g(x)$ is the reflection of $f(x)$ over both axes.
$g(x)$ is the reflection of $f(x)$ over the $y$-axis.

Explanation:

Step1: Recall y-axis reflection rule

A reflection of $f(x)$ over the $y$-axis is given by $f(-x)$.

Step2: Substitute $f(x)$ into the rule

Given $f(x)=0.7(6)^x$, then $f(-x)=0.7(6)^{-x}$.

Step3: Compare to $g(x)$

We see $g(x)=0.7(6)^{-x}=f(-x)$, which matches the y-axis reflection definition.

Answer:

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