The real valued function $f(x) = \frac{x}{e^x - 1} + \frac{x}{2} + 1$ defined on $R \backslash \{0\}$ is

Let $A \subseteq R, B \subseteq R$ and $f: A \rightarrow B$ be defined by $f(x)=x^2-3x+2$. If $f$ is a bijection,then

Check the injectivity and surjectivity of the function $f: Z \rightarrow Z$ defined by $f(x) = x^{2}$.

Let $E = \{ 1, 2, 3, 4 \} $ and $F = \{ 1, 2 \} $. Then the number of onto functions from $E$ to $F$ is

Let $A = \{1, 2, 3, \ldots, 7\}$ and let $P(A)$ denote the power set of $A$. If the number of functions $f: A \rightarrow P(A)$ such that $a \in f(a)$ for all $a \in A$ is $m^n$,where $m, n \in N$ and $m$ is the least possible value,then $m + n$ is equal to . . . . . . .

For $A = \{-1, -2, 3, 4\}$,the number of one-one functions from $A$ to $A$ is . . . . . . .