The function $f(x) = \sin (\log (x + \sqrt {x^2 + 1}))$ is

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

Let $R$ be the set of real numbers and $f: R \rightarrow R$ be defined by $f(x) = \frac{\{x\}}{1+[x]^2}$,where $[x]$ is the greatest integer less than or equal to $x$,and $\{x\} = x-[x]$. Which of the following statements are true?
$I.$ The range of $f$ is a closed interval.
$II.$ $f$ is continuous on $R$.
$III.$ $f$ is one-one on $R$.

If the function $f: R \rightarrow R$ is defined by $f(x)=x|x|$,then:

Which of the following function$(s)$ is/are transcendental?

Let $x$ denote the total number of one-one functions from a set $A$ with $3$ elements to a set $B$ with $5$ elements,and $y$ denote the total number of one-one functions from the set $A$ to the set $A \times B$. Then ...... .