Let $f(x) = \left[ {2{x^2} + 1} \right]$ and $$g(x) = \left\{ {\matrix{ {2x - 3,} & {x < 0} \cr {2x + 3,} & {x \ge 0} \cr } } \right.$$, where [t] is the greatest integer $\le$ t. Then, in the open interval ($-$1, 1), the number of points where fog is discontinuous is equal to ______________.
Answer (integer)
62
Solution
$\mathrm{f}(\mathrm{g}(\mathrm{x}))=\left[2 \mathrm{~g}^2(\mathrm{x})\right]+1$<br/><br/>
$$
=\left\{\begin{array}{l}
{\left[2(2 x-3)^2\right]+1 ; x<0} \\
{\left[2(2 x+3)^2\right]+1 ; x \geq 0}
\end{array}\right.
$$<br/><br/>
$\therefore$ fog is discontinuous whenever $2(2 x-3)^2$ or $2(2 x+3)^2$ belongs to integer except $x=0$<br/><br/>
$\therefore 62$ points of discontinuity.
About this question
Subject: Mathematics · Chapter: Limits, Continuity and Differentiability · Topic: Limits and Standard Results
This question is part of PrepWiser's free JEE Main question bank. 162 more solved questions on Limits, Continuity and Differentiability are available — start with the harder ones if your accuracy is >70%.