Let $f : R \to R$ be defined as $f(x) = e^{x^2} + \cos x$. Then $f$ is:

If $f: R \rightarrow R$ is defined by $f(x)=x-[x]-\frac{1}{2}$ for $x \in R$, where $[x]$ is the greatest integer not exceeding $x$, then $\{x \in R: f(x)=\frac{1}{2}\}$ is equal to :

If $P(S)$ denotes the set of all subsets of a given set $S$,then the number of one-to-one functions from the set $S = \{ 1, 2, 3 \}$ to the set $P(S)$ is

Let $g: N \rightarrow N$ be defined as
$g(3n+1)=3n+2$
$g(3n+2)=3n+3$
$g(3n+3)=3n+1, \text{ for all } n \geq 0$
Then which of the following statements is true?

Let $N$ be the set of all natural numbers,$Z$ be the set of all integers and $\sigma: N \rightarrow Z$ be defined by $\sigma(n)=\begin{cases} \frac{n}{2}, & \text{if } n \text{ is even} \\ -\frac{n-1}{2}, & \text{if } n \text{ is odd} \end{cases}$. Then,

If a function $f: Z \rightarrow Z$ is defined by $f(x) = x - (-1)^x$,then $f(x)$ is