Question

Let $f ( x )=\left[2 x ^{2}+1\right]$ and $g ( x )=\begin{cases}2 x -3, & x <0 \\ 2 x +3, & x \geq 0\end{cases},$, where $[ t ]$ is the greatest integer $\leq t$. Then, in the open interval $(-1,1)$, the number of points where fog is discontinuous is equal to_____

Answer 1

$f ( g ( x ))=\left[2 g ^{2}( x )\right]+1$
$=\begin{cases} {\left[2(2 x -3)^{2}\right]+1 ; x < 0} \\ {\left[2(2 x +3)^{2}\right]+1 ; x \geq 0} \end{cases}$
$\therefore$ fog is discontinuous whenever $2(2 x-3)^{2}$ or $2(2 x+3)^{2}$ belongs to integer except $x=0$.
$\therefore 62$ points of discontinuity.