Solve the following for $x,$ where $\left|x\right|$ is modulus function, $\left[x\right]$ is greatest integer function, $\left\{x\right\}$ is a fractional part function.

$\left[x^2\right]-5\left[x\right]+6=0$

Solve the following for $x,$ where $\left|x\right|$ is modulus function, $\left[x\right]$ is greatest integer function, $\left\{x\right\}$ is a fractional part function.

$[x-2]+[x+2]+\left\{x\right\}=0$

Solve the following for $x,$ where $\left|x\right|$ is modulus function, $\left[x\right]$ is greatest integer function, $\left\{x\right\}$ is a fractional part function.

$\left[\frac{x}{2}\right]+\left[\frac{x}{3}\right]=\frac{5x}{6}$

Find the domain of the following function.

$f\left(x\right)=\frac{x^2+4x+4}{x^2+x-6}$

Find the domain of the following function.

$f\left(x\right)=\sqrt{x-3}+\frac{1}{\log (5-x)}$

Find the domain of the following function.

$f\left(x\right)=\sqrt{1-\sqrt{1-\sqrt{1-x^2}}}$

Find the domain of the following function.
$f\left(x\right)=x !$

Find the domain of the following function.

$f\left(x\right){=}^{5-x}{\mathrm{P}}_{x-1}$

Find the domain of the following function.

$f\left(x\right)=\sqrt{x-x^2}+\sqrt{5-x}$