Let $f: R \rightarrow R$ be a function defined by $f(x) = \frac{e^{|x|} - e^{-x}}{e^x + e^{-x}}$, then

Let $Z$ denote the set of integers. Define $f: Z \rightarrow Z$ by $f(x) = \begin{cases} \frac{x}{2}, & x \text{ is even} \\ 0, & x \text{ is odd} \end{cases}$. Then $f$ is:

The number of functions $f:[0,1] \rightarrow [0,1]$ satisfying $|f(x)-f(y)|=|x-y|$ for all $x, y \in [0,1]$ is

For functions $f$ and $g$,where $f: [0, \frac{\pi}{2}] \rightarrow R$ with $f(x) = \sin x$ and $g: [0, \frac{\pi}{2}] \rightarrow R$ with $g(x) = \cos x$,which of the following is true?

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,

Let $S = \{1, 2, 3, 4, 5, 6\}$. Then the number of one-one functions $f: S \rightarrow P(S)$,where $P(S)$ denotes the power set of $S$,such that $f(n) \subset f(m)$ whenever $n < m$ is $..................$