Find the inverse function and determine if the relation is a function. Also, find the equation for g(x) and its inverse.

We can also write 8 as $2^3$, so $a = (2^3)^{1/4} = 2^{3/4}$.

This can also be written as $y = 8^{x/4}$.
The equation for $g(x)$ is $\boxed{y = 8^{x/4}}$.

2.2.2
To determine the equation of $g^{-1}(x)$, we first swap $x$ and $y$ in the equation for $g(x)$. The question asks for the answer in the form $x = ...$.

The equation of $g^{-1}(x)$ in the form $x = ...$ is $\boxed{x = 8^{y/4}}$.

2.2.3
The function $g(x) = 8^{x/4}$ is an exponential function. Exponential functions of the form $y = b^x$ have a horizontal asymptote at $y=0$.
The inverse function $g^{-1}(x)$ is a logarithmic function. The asymptote of an inverse function is found by swapping the variables of the original function's asymptote.
Since $g(x)$ has a horizontal asymptote at $y=0$, its inverse $g^{-1}(x)$ will have a vertical asymptote at $x=0$.
The equation for the asymptote of $g^{-1}(x)$ is $\boxed{x = 0}$.