The number of strictly increasing functions $f$ from the set $A = \{1, 2, 3, 4, 5, 6\}$ to the set $B = \{1, 2, 3, \dots, 9\}$ such that $f(i) \neq i$ for all $1 \le i \le 6$ is equal to:

If $n(A) = 5$ and $n(B) = 8$,how many possible functions can be defined from $A$ to $B$?

If $f: \{1, 2, 3, 4\} \to \{1, 2, 3, 4\}$ is a function such that $|f(\alpha) - \alpha| \leqslant 1$ for all $\alpha \in \{1, 2, 3, 4\}$,then the total number of such functions is:

Show that the function $f: R \rightarrow R$ defined as $f(x) = x^{2}$ is neither one-one nor onto.

If $f(x) = [8x] - 3$,where $[x]$ is the greatest integer function of $x$,then $f(\pi) = $

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 $R$.