There is one way, which is to use the fact that $ \ f(g(g^{-1}(x)))=f(x)$. But this method only works if $g$ has a right inverse. There are other heuristic methods, which is to "guess the shape" of $ \ f$, given the composite function.
But is there a more powerful method that can be profitably used by non-calculus students? What about methods from calculus?
I ask this because the Malaysian public exams ask these questions a whole lot, and students normally jump right into these so-called "find the outside function" questions by assuming that $g$ is invertible, and then get into a knot when it's not.
You can try letting $y=g(x)$, and finding an expression for $f(y)$.
Example:
$fg(x)=x^2+2$, $g(x)=x-1$.
Let $y=x-1$. Hence $x=1+y$. Thus $fg(x)=f(y)=(1+y)^2+2=y^2+2y+3$.
Thus $f(x)=x^2+2x+3$.