The function $f: R \rightarrow R$ defined by $f(x)=\frac{x}{\sqrt{1+x^2}}$ is

Let $R$ be the set of all real numbers and $f: R \rightarrow R$ be a continuous function. Suppose $|f(x) - f(y)| \geq |x - y|$ for all real numbers $x$ and $y$. Then,

Let $f: R \to R$ be defined as $f(x) = x^3$. Then $f$ is . . . . . . .

If $f: R \rightarrow R$ is defined by $f(x) = x^2 + 3x + 4$,then the function $f$ is . . . . . . .

Let $X$ be a set with exactly $5$ elements and $Y$ be a set with exactly $7$ elements. If $\alpha$ is the number of one-one functions from $X$ to $Y$ and $\beta$ is the number of onto functions from $Y$ to $X$,then the value of $\frac{1}{5!}(\beta-\alpha)$ is.

Let $f :(0,1) \rightarrow R$ be a function defined by $f(x)=\frac{1}{1-e^{-x}}$,and $g(x)=(f(-x)-f(x))$. Consider two statements:
$(I)$ $g$ is an increasing function in $(0,1)$
$(II)$ $g$ is one-one in $(0,1)$
Then,