Answer
$f(x)=e^x$
$g(x)=\ln x$
Work Step by Step
Since we have to determine two different functions $f$ and $g$ so that $f(g(x))=g(f(x))$, the easiest way is to consider a one-to-one function $f$ and find its inverse $g$.
For example:
$$f(x)=e^x\text{ and }g(x)=\ln x.$$
Check if the two functions verify the conditions:
$$\begin{align*}
f(g(x))&=e^{\ln x}=x\\
g(f(x))&=\ln (e^x)=x\ln e=x.
\end{align*}$$
Since $f(g(x))=x=g(f(x))$ and $f(x)\not=g(x)$, the two functions check the conditions.
