Consider two sets $A = \{ x \in \mathbb{Z} : |(| x - 3| - 3)| \leq 1 \}$ and $B = \{ x \in \mathbb{R} - \{1, 2\} : \frac{(x - 2)(x - 4)}{x - 1} \log_{e}(|x - 2|) = 0 \}$. Then the number of onto functions $f: A \rightarrow B$ is equal to:

Which one of the following is not bounded on the intervals as indicated?

If $f(x) = \log_{e}\left(\frac{1-x}{1+x}\right)$,$|x| < 1$,then $f\left(\frac{2x}{1+x^2}\right)$ is equal to

Let $N$ be the set of positive integers. For all $n \in N$,let $f_n = (n+1)^{1/3} - n^{1/3}$ and $A = \{n \in N : f_{n+1} < \frac{1}{3(n+1)^{2/3}} < f_n\}$. Then,

If $f(x)=\frac{2^{x}+2^{-x}}{2}$,then $f(x+y) \cdot f(x-y)$ is

If the set of all values of $a$ is $[\alpha, \beta] \cup [\gamma, \delta]$ for which the function $f(x) = \begin{cases} 3x + |a^2 - 4|; & a \leqslant x < 1 \\ 5 - x^2; & x \geqslant 1 \end{cases}$ has its largest value at $x = 1$,then find the value of $(\alpha + \beta + \gamma + \delta)$.