Let $\alpha, \beta$ and $\gamma$ be three positive real numbers. Let $f(x) = \alpha x^5 + \beta x^3 + \gamma x, x \in \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be such that $g(f(x)) = x$ for all $x \in \mathbb{R}$. If $a_1, a_2, a_3, \dots, a_n$ are in arithmetic progression with mean zero,then the value of $f(g(\frac{1}{n} \sum_{i=1}^{n} f(a_i)))$ is equal to.

Let $[x]$ denote the integral part of $x \in R$. Let $g(x) = x - [x]$. Let $f(x)$ be any continuous function with $f(0) = f(1)$. Then the function $h(x) = f(g(x))$:

Let $f(x)$ and $g(x)$ be two continuous functions defined from $R \rightarrow R$,such that $f(x_1) > f(x_2)$ and $g(x_1) < g(x_2)$ for all $x_1 > x_2$. Then the solution set of $f(g(\alpha^2 - 2\alpha)) > f(g(3\alpha - 4))$ is

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 graph of a non-constant function $f(x)$ is symmetric about the point $(3,4)$,then the value of $\sum\limits_{r = 0}^6 {f(r) + f(3)}$ is equal to

If $f(x)=2\{x\}+5x$,where $\{x\}$ is the fractional part function,then $f(-1.4)$ is