Range of the function

$f(x) = \sqrt {\left| {{{\sin }^{ – 1}}\left| {\sin x} \right|} \right| – {{\cos }^{ – 1}}\left| {\cos x} \right|} $ is

If $h\left( x \right) = \left[ {\ln \frac{x}{e}} \right] + \left[ {\ln \frac{e}{x}} \right]$ ,where [.] denotes greatest integer function, then which of the following is false ?

Suppose that a function $f: R \rightarrow R$ satisfies $f(x+y)=f(x) f(y)$ for all $x, y \in R$ and $f(1)=3 .$ If $\sum \limits_{i=1}^{n} f(i)=363,$ then $n$ is equal to

Let $f(x) = sin\,x,\,\,g(x) = x.$

Statement $1:$ $f(x)\, \le \,g\,(x)$ for $x$ in $(0,\infty )$

Statement $2:$ $f(x)\, \le \,1$ for $(x)$ in $(0,\infty )$ but $g(x)\,\to \infty$ as $x\,\to \infty$

If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $P ( x )=f\left( x ^{3}\right)+ xg \left( x ^{3}\right)$ is divisible by $x^{2}+x+1,$ then $P(1)$ is equal to ……. .