If a function $g(x)$ is defined in $[-1, 1]$ and two vertices of an equilateral triangle are $(0, 0)$ and $(x, g(x))$ and its area is $\frac{\sqrt{3}}{4}$,then $g(x)$ equals :-

If $f(x)$ is a real-valued function defined by $f(x) = \frac{a x^{10} + b x^8 + c x^6 + d x^4 + e x^2 + 12 x + 15}{x}$ for $x \neq 0$ and $f(4) = -4$, then find the value of $f(-4)$.

If $f(x) = \cos (\log x)$,then $f(x)f(y) - \frac{1}{2}[f(x/y) + f(xy)] = $

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 $f : R \rightarrow R$ 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 $f(x) = Ax^3 - Bx - \tan x \cdot \text{sgn}(x)$ be an even function $\forall x \in R - \left\{ (2n + 1)\frac{\pi}{2}, n \in I \right\}$,where $A = \sin^2 \alpha - \sin \alpha + \frac{1}{4}$ and $B = \tan^2 \alpha + \frac{2}{\sqrt{3}} \tan \alpha + \frac{1}{3}$. Then the number of values of $\alpha$ in $\left[ -\frac{3\pi}{2}, 2\pi \right]$ is (where $\text{sgn}(x)$ denotes the signum function of $x$).