The number of one-one functions from the set $A = \{1, 2, 3, 4, 5\}$ to itself is . . . . . . .

If $f: R \rightarrow R$ is defined by $f(x)=2x+\sin x, x \in R$,then $f$ is

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

Let $f: \mathbb{Z} \rightarrow \mathbb{Z}$ be defined by $f(x) = x^3 + 2$. Then,$f$ is . . . . . . .

The number of bijective functions $f : \{1, 3, 5, 7, \ldots, 99\} \rightarrow \{2, 4, 6, 8, \ldots, 100\}$ such that $f(3) \geq f(9) \geq f(15) \geq f(21) \geq \ldots \geq f(99)$ is:

Let $a > 1$ and $0 < b < 1$. If $f: R \rightarrow [0, 1]$ is defined by $f(x) = \begin{cases} a^x, & -\infty < x < 0 \\ b^x, & 0 \leq x < \infty \end{cases}$,then $f(x)$ is