JEE MAIN - Mathematics (2022 - 25th June Evening Shift - No. 16)
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
62
Explanation
$$
\mathrm{f}(\mathrm{g}(\mathrm{x}))=\left[2 \mathrm{~g}^2(\mathrm{x})\right]+1
$$
$$ =\left\{\begin{array}{l} {\left[2(2 x-3)^2\right]+1 ; x<0} \\ {\left[2(2 x+3)^2\right]+1 ; x \geq 0} \end{array}\right. $$
$\therefore$ 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.
$$ =\left\{\begin{array}{l} {\left[2(2 x-3)^2\right]+1 ; x<0} \\ {\left[2(2 x+3)^2\right]+1 ; x \geq 0} \end{array}\right. $$
$\therefore$ 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.
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