Number of integers satisfying the equation $\left[x\right]+\left[2x\right]+\left[3x\right]=\left\{x\right\}+\left\{2x\right\}+2x^2$, (where [.] and $\{.\}$ denotes greatest-integer function and fractional part function respectively) are

Find the domain of the function $\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{\log \left(2-\mathrm{x}\right)}+\sqrt{\mathrm{x}+1}$.

Find the domain of the function $\mathrm{f}\left(\mathrm{x}\right)=\sqrt{1-\mathrm{x}}-{\sin }^{-1}\left(\frac{2\mathrm{x}-1}{3}\right)$.

Find domain and range of following function: $y=x^3$

Find domain and range of the function $\mathrm{y}=\frac{{\mathrm{x}}^2-2\mathrm{x}+5}{{\mathrm{x}}^2+2\mathrm{x}+5}$.