Functions Part 2

The domain of $f\circ g$ consists of all real $x$ for which $g(x)$ is defined and $g(x)>0$, because $f(t)=\log_e t$ requires $t>0$.First check the denominator $2x^2-2x+1$:$2x^2-2x+1=2\left(x-\frac12\right)^2+\frac12>0$ for all real $x$.So $g(x)$ is defined for every real $x$, and its sign is the sign of its numerator.Now simplify the numerator:$x^4-2x^3+3x^2-2x+2=(x^2-x+1)^2+1$.Since $(x^2-x+1)^2\ge 0$, the numerator satisfies $(x^2-x+1)^2+1>0$ for all real $x$.Thus $g(x)>0$ for every real $x$.Therefore the domain of $f\circ g$ is $\mathbb{R}$.Correct option: $\mathbb{R}$, i.e. option C.