We can define composite functions as follows
$(f\cdot g)\left(x\right)=f\left(g\left(x\right)\right)$ $(g\cdot f)\left(x\right)=g\left(f\left(x\right)\right)$
Calculation:
Since $h\left(x\right)=(g\cdot f)\left(x\right)$
Therefore $(g\cdot f)\left(x\right)=\frac{1}{{(x-1)}^{2}}$ $g\left(f\left(x\right)\right)=\frac{1}{{(x-1)}^{2}}$
So, we can assume, $f\left(x\right)=x-1$ and $g\left(x\right)=\frac{1}{{x}^{2}}$
Conclusion:
$f\left(x\right)=x-1$ $g\left(x\right)=\frac{1}{{x}^{2}}$