If $f(x) = \begin{cases} [x] & \text{if } -3 < x \leq -1 \\ |x| & \text{if } -1 < x < 1 \\ |[x]| & \text{if } 1 \leq x < 3 \end{cases}$,then the set $\{x : f(x) \geq 0\}$ is equal to

Let $[\cdot]$ denote the greatest integer function. If $f(x) = [x]$ and $g(x) = 3[\frac{x}{3}]$,then the set of all real $x$ such that $f(x) = g(x)$ is

Let $X$ be a set with exactly $5$ elements and $Y$ be a set with exactly $7$ elements. If $\alpha$ is the number of one-one functions from $X$ to $Y$ and $\beta$ is the number of onto functions from $Y$ to $X$,then the value of $\frac{1}{5!}(\beta-\alpha)$ is.

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

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

The function $f : N \to N$ defined by $f(x) = x - 5[\frac{x}{5}]$,where $N$ is the set of natural numbers and $[x]$ denotes the greatest integer less than or equal to $x$,is