The domain of the function $$f(x)=\sin ^{-1}\left[2 x^{2}-3\right]+\log _{2}\left(\log _{\frac{1}{2}}\left(x^{2}-5 x+5\right)\right)$$, where [t] is the greatest integer function, is :

Let $$f, g: \mathbb{N}-\{1\} \rightarrow \mathbb{N}$$ be functions defined by $$f(a)=\alpha$$, where $$\alpha$$ is the maximum of the powers of those primes $$p$$ such that $$p^{\alpha}$$ divides $$a$$, and $$g(a)=a+1$$, for all $$a \in \mathbb{N}-\{1\}$$. Then, the function $$f+g$$ is

If the maximum value of $$a$$, for which the function $$f_{a}(x)=\tan ^{-1} 2 x-3 a x+7$$ is non-decreasing in $$\left(-\frac{\pi}{6}, \frac{\pi}{6}\right)$$, is $$\bar{a}$$, then $$f_{\bar{a}}\left(\frac{\pi}{8}\right)$$ is equal to

Let f : R $$\to$$ R be a continuous function such that $$f(3x) - f(x) = x$$. If $$f(8) = 7$$, then $$f(14)$$ is equal to :