The function $f(x) = \sec \left[ \log \left( x + \sqrt{1 + x^2} \right) \right]$ is

The function $f: R \rightarrow R$ is defined by $f(x)=3^{-x}$. Observe the following statements about it:
$I$. $f$ is one-one
$II$. $f$ is onto
$III$. $f$ is a decreasing function
Out of these, the true statements are:

Let $f: R \rightarrow R$ be defined by $f(x) = 5^{-|x|} + \operatorname{sgn}(5^{-x})$, where $\operatorname{sgn}(x)$ denotes the signum function of $x$. Then $f$ is

Given that for any $n \in N$, there exist an odd integer $q$ and a non-negative integer $r$ such that $n$ can be written uniquely as $n = q \times 2^r$. Let $f: N \rightarrow N \times N$ be a function defined by $f(n) = \left(r+1, \frac{q+1}{2}\right)$. Then,

In each of the following cases,state whether the function is one-one,onto or bijective. Justify your answer. $f : R \rightarrow R$ defined by $f(x) = 1 + x^2$.

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. . . . . .