If $f: R \rightarrow R$ and $g: R \rightarrow R$ are given by $f(x)=|x|$ and $g(x)=[x]$ for each $x \in R$, then $\{x \in R: g(f(x)) \leq f(g(x))\}$ is equal to

The function $f(x) = |px - q| + r|x|$,$x \in (-\infty, \infty)$,where $p > 0, q > 0, r > 0$ assumes its minimum value only at one point,if

Suppose $f:[-2, 2] \to R$ is defined by $f(x) = \begin{cases} -1 & \text{for } -2 \le x \le 0 \\ x - 1 & \text{for } 0 < x \le 2 \end{cases}$,then find the set $\{ x \in (-2, 2) : x \le 0 \text{ and } f(|x|) = x \}$.

The graph of the function $f(x) = x + \frac{1}{8} \sin(2 \pi x)$,$0 \leq x \leq 1$ is shown below. Define $f_1(x) = f(x)$,$f_{n+1}(x) = f(f_n(x))$,for $n \geq 1$.
Which of the following statements are true?
$I.$ There are infinitely many $x \in [0, 1]$ for which $\lim_{n \rightarrow \infty} f_n(x) = 0$
$II.$ There are infinitely many $x \in [0, 1]$ for which $\lim_{n \rightarrow \infty} f_n(x) = \frac{1}{2}$
$III.$ There are infinitely many $x \in [0, 1]$ for which $\lim_{n \rightarrow \infty} f_n(x) = 1$
$IV.$ There are infinitely many $x \in [0, 1]$ for which $\lim_{n \rightarrow \infty} f_n(x)$ does not exist.

If the set of all values of $a$ is $[\alpha, \beta] \cup [\gamma, \delta]$ for which the function $f(x) = \begin{cases} 3x + |a^2 - 4|; & a \leqslant x < 1 \\ 5 - x^2; & x \geqslant 1 \end{cases}$ has its largest value at $x = 1$,then find the value of $(\alpha + \beta + \gamma + \delta)$.

Let $R = \{(1, 3), (2, 2), (3, 2)\}$ and $S = \{(2, 1), (3, 2), (2, 3)\}$ be two relations on set $A = \{1, 2, 3\}$. Then $RoS =$