Let \(f\left(x\right)={\log }_{e}x\text{ and }g\left(x\right)=\frac{{x}^{4}-2{x}^{3}+3{x}^{2}-2x+2}{2{x}^{2}-2x+1}\). Th…

\(\text{Given, }\\ f\left(x\right)={\log }_{e}x\\ \text{then}{D}_{f}\in \left(0,\infty \right)\\ \text{and}g\left(x\right)=\frac{{x}^{4}-2{x}^{3}+3{x}^{2}-2x+2}{2{x}^{2}-2x+1}\\ \text{then}{D}_{g}\in R\\ \text{Now, for}fog\left(x\right)=f\left(g\left(x\right)\right)\\ ={\log }_{e}\left(\frac{{x}^{4}-2{x}^{3}+3{x}^{2}-2x+2}{2{x}^{2}-2x+1}\right)\\ \text{Now, Domain for}fog\left(x\right)\\ \frac{{x}^{4}-2{x}^{3}+3{x}^{2}-2x+2}{2{x}^{2}-2x+1}>0\forall x\in R\\ \text{Therefore }{\log }_{e}\left(\frac{{x}^{4}-2{x}^{3}+3{x}^{2}-2x+2}{2{x}^{2}-2x+1}\right)\text{ is defined ∀}x\in R\\ \\\)