Which one of the following is not bounded on the intervals as indicated?

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 $f: R \rightarrow [-1, 1]$ and $g: R \rightarrow A$ are two surjective mappings and $\sin \left(g(x) - \frac{\pi}{3}\right) = \frac{f(x)}{2} \sqrt{4 - f^2(x)}$,then $A =$

Let $R$ denote the set of all real numbers. Let $f: R \rightarrow R$ be a function such that $f(x) > 0$ for all $x \in R$,and $f(x+y)=f(x) f(y)$ for all $x, y \in R$. Let the real numbers $a_1, a_2, \ldots, a_{50}$ be in an arithmetic progression. If $f(a_{31})=64 f(a_{25})$,and $\sum_{i=1}^{50} f(a_i)=3(2^{25}+1)$,then the value of $\sum_{i=6}^{30} f(a_i)$ is:

If $f(x) = 2x$ and $g$ is the identity function,then:

Let the function $f:[0,1] \rightarrow \mathbb{R}$ be defined by $f(x) = \frac{4^x}{4^x+2}$. Then the value of $f\left(\frac{1}{40}\right) + f\left(\frac{2}{40}\right) + f\left(\frac{3}{40}\right) + \dots + f\left(\frac{39}{40}\right) - f\left(\frac{1}{2}\right)$ is: