find the domain of these functions.please explain them briefly.

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{Given}, \\ \mathrm{f}\left(\mathrm{x}\right)=\sqrt{{\mathrm{x}}^{12}-{\mathrm{x}}^9+{\mathrm{x}}^4-\mathrm{x}+1} \\ {\mathrm{x}}^{12}-{\mathrm{x}}^9+{\mathrm{x}}^4-\mathrm{x}+1>0 \left(\mathrm{since} \mathrm{it} \mathrm{lies} \mathrm{inside} \mathrm{root}\right) \\ {\mathrm{x}}^{12}+{\mathrm{x}}^4-{\mathrm{x}}^9-\mathrm{x}+1>0 \\ {\mathrm{x}}^4\left({\mathrm{x}}^8+1\right)-\mathrm{x}\left({\mathrm{x}}^8+1\right)+1>0 \\ \left({\mathrm{x}}^8+1\right)\left({\mathrm{x}}^4-\mathrm{x}\right)+1>0 \\ \mathrm{now},\left({\mathrm{x}}^8+1\right) \mathrm{will} \mathrm{be} >0 \mathrm{for} \mathrm{all} \mathrm{x} \\ \mathrm{taking} \left({\mathrm{x}}^4-\mathrm{x}\right) \\ \mathrm{for} \mathrm{x}\le 0 \mathrm{and} \mathrm{x}\ge 1 \\ {\mathrm{x}}^4\ge \mathrm{x} \\ \mathrm{hence} \left({\mathrm{x}}^4-\mathrm{x}\right)\ge 0 \\ \mathrm{hence} \left({\mathrm{x}}^8+1\right)\left({\mathrm{x}}^4-\mathrm{x}\right)+1>0 \mathrm{will} \mathrm{be} \mathrm{true} \mathrm{for} \left( \mathrm{x}\le 0 \mathrm{and} \mathrm{x}\ge 1\right) \\ \mathrm{for} \mathrm{x}\in \left(0,1\right) \\ {\mathrm{x}}^4<\mathrm{x} \\ {\mathrm{x}}^4-\mathrm{x}<0 \\ \mathrm{since} 0< \mathrm{x}<1 \\ {\mathrm{x}}^8<<<<<<1 \\ \mathrm{hence} \mathrm{we} \mathrm{can} \mathrm{approximate} {\mathrm{x}}^8+1 \mathrm{to} 1 \\ \mathrm{hence} \mathrm{we} \mathrm{have} 1.\left({\mathrm{x}}^4-\mathrm{x}\right)+1>0 \\ \mathrm{since},-1< \left({\mathrm{x}}^4-\mathrm{x}\right)<0 \left(\mathrm{x}\in \left(0,1\right)\right) \\ \mathrm{hence} \mathrm{domain} \mathrm{of} f\left(\mathrm{x}\right) \mathrm{will} \mathrm{be} \mathrm{R} \\ \mathbf{x}\mathbf{\in }\mathbf{R} \end{gathered}$$


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