Subject: Mathematics — Trigonometry
Question: For a real number $\alpha$, let $\lbrack\alpha\rbrack$
denote the greatest integer less than or equal to $\alpha$. For a finite
set
S, let $|S|$ denote the number of elements in the set S.
Consider the functions $f:( - 3,3) \rightarrow ( - \infty,\infty)$ and
$g:( - 3,3) \rightarrow ( - \infty,\infty)$ defined by
$$f(x) = \left\lbrack x^{3} \right\rbrack\log_{e}\left( 1 + \sin^{2}(\pi(x - \lbrack x\rbrack)) \right)
$$and
$(x) = x^{3}\sin^{2}\left( \pi\log_{e}(1 + x - \lbrack x\rbrack) \right)$.\
Let $A = \{ x \in ( - 3,3):f$ is discontinuous at x$\}$\
and $B = \{ x \in ( - 3,3):g$ is discontinuous at x$\}$.\
Then the value of $|A| + 2|\text{ }B| - |A \cap B|$ is $\_\_\_\_$ .
Answer: 56.00
Solution: $f(x) = \left\lbrack x^{3} \right\rbrack log\left( 1 + \sin^{2}\pi(x - \lbrack x\rbrack) \right)$\
$${f(x) = \left\lbrack x^{3} \right\rbrack log\left( 1 + \sin^{2}\pi x \right)
}{x \in ( - 3,3)
}{x^{3} \in ( - 27,27)
}$$At integers $x = \pm 1, \pm 2,0$\
$f(x)$ is continuous as $log\left( 1 + \sin^{2}\pi x \right) = 0$\
So number of points of discontinuity of $f(x)$ is $53 - 5 = 48$\
and $g(x) = x^{3}\sin^{2}\pi\log_{e}(1 + \{ x\})$ is cont. at $x = 0$,\
So point of Discontinuity are $x = \pm 1\& \pm 2$\
$${|A| = 48\&|\text{ }B| = 4\&|\text{ }A \cap \text{ }B| = 0
}{|A| + 2|\text{ }B| - |A \cap B|
}{= 48 + 8 - 0 = 56}$$