2021-09-18

An expression for f(x) and g(x):

firmablogF

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

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