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 $f(x) = \sqrt{x}$ and $g(x) = x$ be two functions defined over the set of non-negative real numbers. Find $(f+g)(x)$,$(f-g)(x)$,$(fg)(x)$,and $(\frac{f}{g})(x)$.

Let $[x]$ represent the greatest integer less than or equal to $x$,${x} = x - [x]$,$\sqrt{2} = 1.414$ and $\sqrt{3} = 1.732$. If $f(x) = \{x + [\frac{x}{1+x^2}]\}$ is a real-valued function,then $f(\sqrt{2}) + f(-\sqrt{3}) = $

Let $N$ be the set of all natural numbers and $f: N \rightarrow N$ be such that $1990 < f(1990) < 2100$ and satisfies the equation $x-f(x)=19[\frac{x}{19}]-90[\frac{f(x)}{90}]$,where $[y]$ denotes the greatest integer less than or equal to $y$. Then the number of possible values of $f(1990)$ is

Let $f(x) = \frac{\sqrt{x - 2\sqrt{x - 1}}}{\sqrt{x - 1} - 1}$. Then:

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.