The domain of the function $f(x) = e^{\sqrt{5x - 3 - 2x^2}}$ is

Let $R$ be the relation on $Z$ defined by $R = \{(a, b) : a, b \in Z, a - b \text{ is an integer}\}$. Find the domain and range of $R$.

If the domain of the function $\log _e\left(\frac{6 x^2+5 x+1}{2 x-1}\right)+\cos ^{-1}\left(\frac{2 x^2-3 x+4}{3 x-5}\right)$ is $(\alpha, \beta) \cup(\gamma, \delta]$,then $18\left(\alpha^2+\beta^2+\gamma^2+\delta^2\right)$ is equal to $....$.

If function $f : R \to S, f(x) = (\sin x - \sqrt{3} \cos x + 1)$ is onto,then $S$ is equal to

Which of the following intervals is a possible domain of the function $f(x) = \log_{\{x\}}[x] + \log_{[x]}\{x\}$,where $[x]$ is the greatest integer not exceeding $x$ and $\{x\} = x - [x]$?

The domain of the function $f(x) = \frac{1}{\log_{10}(1-x)} + \sqrt{x+2}$ is