Let $f(x) = -1 + |x - 2|$ and $g(x) = 1 - |x|$. Then the set of all points where $fog$ is discontinuous is

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 $x$).

Let $f(x)$ be a non-constant polynomial with real coefficients such that $f\left(\frac{1}{2}\right)=100$ and $f(x) \leq 100$ for all real $x$. Which of the following statements is $NOT$ necessarily true?

If $f(x) = \log_e \left( \frac{1-x}{1+x} \right)$,$|x| < 1$,then $f\left( \frac{2x}{1+x^2} \right)$ is equal to

Suppose $f:[-2,2] \rightarrow R$ is defined by $f(x) = \begin{cases} -1, & \text{for } -2 \leq x \leq 0 \\ x-1, & \text{for } 0 < x \leq 2 \end{cases}$. The set $\{x \in [-2,2] : x \leq 0 \text{ and } f(|x|) = x\}$ is equal to:

If $h(x) = [\ln(x/e)] + [\ln(e/x)]$,where $[.]$ denotes the greatest integer function,then which of the following is false?