The exhaustive domain of the following function is $f\left(x\right)=\sqrt{x^{12}-x^9+x^4-x+1}$

$f\left(x\right)=\sqrt{x^{12}-x^9+x^4-x+1}$

We must have $x^{12}-x^9+x^4-x+1\ge 0 \left(1\right)$

Obviously, (1) is satisfied by $x\in \left(-\infty ,0\right].$

$\begin{array}{l}\mathrm{Also}, x^9\left(x^3-1\right)+x\left(x^3-1\right)+1\ge 0\forall x\in \left[1,\infty \right).\\ \mathrm{Further}, x^{12}-x^9+x^4-x+1=\left(1-x\right)+x^4\left(1-x^5\right)+x^{12} \mathrm{is} \mathrm{also}\\ \mathrm{satisfied} \mathrm{by} x\in \left(0,1\right).\\ \mathrm{Hence}, \mathrm{the} \mathrm{domain} \mathrm{is}R.\end{array}$