Range of the function , $f (x) = cot ^{-1}$ $\left( {{{\log }_{4/5}}\,\,(5\,{x^2}\,\, – \,\,8\,x\,\, + \,\,4)\,} \right)$ is :

Let $f, g: N -\{1\} \rightarrow N$ 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 $a \in N -\{1\}$. Then, the function $f+ g$ is.

Let $N$ be the set of positive integers. For all $n \in N$, let $f_n=(n+1)^{1 / 3}-n^{1 / 3} \text { and }$ $A=\left\{n \in N: f_{n+1}<\frac{1}{3(n+1)^{2 / 3}} < f_n\right\}$ Then,

If the function $f\,:\,R – \,\{ 1, – 1\}  \to A$ defined by $f\,(x)\, = \frac{{{x^2}}}{{1 – {x^2}}},$ is surjective, then $A$ is equal to

Let, $f(x)=\left\{\begin{array}{l} x \sin \left(\frac{1}{x}\right) \text { when } x \neq 0 \\ 1 \text { when } x=0 \end{array}\right\}$ and $A=\{x \in R: f(x)=1\} .$ Then, $A$ has