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

Let $f : R \rightarrow R$ and $g : R \rightarrow R$ be two functions defined by $f(x)=\log _{e}(x^{2}+1)-e^{-x}+1$ and $g(x)=\frac{1-2e^{2x}}{e^{x}}$. Then,for which of the following range of $\alpha$,the inequality $f(g(\frac{(\alpha-1)^{2}}{3})) > f(g(\alpha-\frac{5}{3}))$ holds?

Let $f:[0,2] \rightarrow R$ be the function defined by $f(x)=(3-\sin(2\pi x)) \sin(\pi x-\frac{\pi}{4})-\sin(3\pi x+\frac{\pi}{4})$. If $\alpha, \beta \in[0,2]$ are such that $\{x \in[0,2]: f(x) \geq 0\}=[\alpha, \beta]$,then the value of $\beta-\alpha$ is:

Let $f(x) = \begin{cases} x-1, & x \text{ is even} \\ 2x, & x \text{ is odd} \end{cases}$. If for some $a \in N, f(f(f(a))) = 21$,then $\lim_{x \rightarrow a^{-}} \left\{ \frac{|x|^3}{a} - \left[ \frac{x}{a} \right] \right\}$,where $[t]$ denotes the greatest integer less than or equal to $t$,is equal to:

Let $f : R \to R$ be a function defined by $f(x) = - \frac{|x|^3 + |x|}{1 + x^2}$; then the graph of $f(x)$ lies in the :-

The number of roots of the equation $\sqrt{2}+e^{\cosh^{-1} x}-e^{\sinh^{-1} x}=0$ is