If the function $f: R \rightarrow R$ is defined by $f(x) = |x|(x - \sin x)$,then which of the following statements is $TRUE$?

Let $f: R \rightarrow R$ be defined by $f(x) = x^{2} - \frac{x^{2}}{1+x^{2}}$ for all $x \in R$. Then,

The function $f(x) = x^{2} + bx + c$, where $b$ and $c$ are real constants, describes:

Let $R$ be the set of all real numbers. Statement $I$: The function $f: \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \rightarrow R$ defined by $f(x) = \sec x + \tan x$ is a one-one function. Statement $II$: The function $f: [0, \infty) \rightarrow R$ defined by $f(x) = x^2$ is a one-one function. Which of the above statements is(are) true?

Show that the function $f: R_* \rightarrow R_*$ defined by $f(x) = \frac{1}{x}$ is one-one and onto,where $R_*$ is the set of all non-zero real numbers. Is the result true if the domain $R_*$ is replaced by $N$ with the co-domain remaining the same as $R_*$?

Let $g(x) = 1 + x - [x]$ and $f(x) = \begin{cases} -1, & x < 0 \\ 0, & x = 0 \\ 1, & x > 0 \end{cases}$. Then for all $x$,$f(g(x))$ is equal to