$f(x)=\frac{x}{e^x-1}+\frac{x}{2}+2 \cos ^3 \frac{x}{2}$ on $R-\{0\}$ is

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 $..................$

The number of functions $f: \{1, 2, \ldots, 100\} \rightarrow \{0, 1\}$ that assign $1$ to exactly one of the positive integers less than or equal to $98$ is equal to $\qquad$.

If $R$ denotes the set of all real numbers,then the function $f: R \rightarrow R$ defined by $f(x)=|x|$ is

Let a function $f: (0, \infty) \to (0, \infty)$ be defined by $f(x) = |1 - \frac{1}{x}|$. Then $f$ is

If $f: R \rightarrow R$ is defined by $f(x) = x^2 + 3x + 4$,then the function $f$ is . . . . . . .