The range of the real valued function $f(x) = \frac{x^2+x+1}{x}$ is

The range of $f(x) = \cos[x]$ for $-\frac{\pi}{4} < x < \frac{\pi}{4}$ (where $[.]$ represents the greatest integer function less than or equal to $x$) is

Find the set $\{x \in R : \frac{\sqrt{|x|^2-2|x|-8}}{\log(2-x-x^2)} \text{ is a real number}\}$.

The range of the function $f(x) = \frac{x}{x^2 - 5x + 9}$ is

Let the domain of the function $f(x) = \cos^{-1}\left(\frac{4x+5}{3x-7}\right)$ be $[\alpha, \beta]$ and the domain of $g(x) = \log_2\left(2-6\log_{27}(2x+5)\right)$ be $(\gamma, \delta)$. Then $|7(\alpha+\beta)+4(\gamma+\delta)|$ is equal to . . . . . . .

If the domain of the function $\log _5(18 x-x^2-77)$ is $(\alpha, \beta)$ and the domain of the function $\log _{(x-1)}\left(\frac{2 x^2+3 x-2}{x^2-3 x-4}\right)$ is $(\gamma, \delta)$,then $\alpha^2+\beta^2+\gamma^2$ is equal to :