Why do these functions satisfy that f(n) is not O(g(n)) and g(n) is not O(f(n))?
Let's firstly imagine function $f$: for $k=2$ we have two intervals $(4,5)$ and $(5,6)$ and for $k=3$ we have two intervals $(6,7)$ and $(7,8)$. Accordingly for $f$ we have:
$$\quad\quad\quad   x\quad\ \overbrace{(4,5)\quad (5,6)}^{k=2}\quad\overbrace{(6,7)\quad (7,8)}^{k=3} \cdots\\
f(x)\quad  2^4\quad\quad 2^5\quad\quad\quad 3^6 \quad\quad3^7$$
for function $g$ we have:
$$\quad\quad\quad   x\quad\ \overbrace{(4,5)\quad (5,6)}^{k=2}\quad\overbrace{(6,7)\quad (7,8)}^{k=3} \cdots\\
g(x)\quad  2^2\quad\quad 2^6\quad\quad\quad 3^4 \quad\quad3^8$$
So, as you see both $f$ and $g$ are increasing, but alternately one is more then  other with more then constant factor, so they cannot be big-$O$ of each other.
To get an idea of an example you can take a look at DNA molecule chain  and imagine, that $f$ is one line and $g$ another.