Domains and Ranges of Common Functions

Let $f\left(x\right)=\frac{\sin x}{\sqrt{1+{\tan }^{2}x}}-\frac{\cos x}{\sqrt{1+{\cot }^{2}x}}$. Then range of $f\left(x\right)$ is

The least value of $x$ for which $f\left(x\right)=\frac{\sqrt{x^2-4}}{{\sin }^{-1}(2-x)}$ is defined is

The domain of $f\left(x\right)=\sqrt{\cos \left(\sin x\right)}+\sqrt{{\log }_x\left\{x\right\}}; \left\{·\right\}$ denotes the fractional part, is

Let $f$ be a function with domain $\left[-3,5\right]$ and let $g\left(x\right)=\left|3x+4\right|$. Then the domain of $\left(fog\right)\left(x\right)$ is

The range of the function $f\left(x\right)=\frac{5}{3-x^2}$ is

The set of all real numbers $x$ for which ${\log }_{2004}\left({\log }_{2003}\left({\log }_{2002}\left({\log }_{2001}x\right)\right)\right)$ is defined as $\{x\mid x>c\}.$ The value of $c$ is

The domain of the function ${\cos }^{-1}{\log }_2\left(x^2+5x+8\right)$ is

If $f\left(x\right)$ is defined on domain $\left[0,1\right]$, then $f\left(2\sin x\right)$ is defined on

The range of $f\left(x\right)=\cos \frac{\pi \left[x\right]}{2}$ where $\left[·\right]$ denotes greatest integer function is

The set of all values of $x$ in $[0,\pi ]$ for which the function $\sqrt{1-2\sin 3x}$ takes real values is

The domain and range of the function $f$ given by $f\left(x\right)=2-\left|x-5\right|$ is

The function $y\left(x\right)$ given implicitly by the equation $3\sin y+5\sin x=1$ is real in

The function $y\left(x\right)$ given implicitly by $3\sin y+a\sin x=1, a>0$ exists provided, then

If $f\left(x\right)=\ln \left(\frac{x^2+e}{x^2+1}\right),$ then range of $f\left(x\right)$ is

Domain of ${\cos }^{-1}\left[2x^2-3\right],$ where $[·]$ denotes greatest integer function is

The domain of the functions $f\left(x\right)={\log }_2{\log }_3{\log }_{4}x$ is

If ${\log }_{\frac{1}{2}}\left(x^2-5x+7\right)>0,$ then exhaustive range of values of $x$ is

The domain of the function $f\left(x\right)=\frac{1}{\sqrt{x^2-3x+2}}$ is