Question

Two functions $f\left(x\right)$ and $g\left(x\right)$ are defined as $f\left(x\right)={\log }_{10}\left|\frac{x-2}{x^2-10x+24}\right|$ and $g\left(x\right)={\sin }^{-1}\left(\frac{2\left[x\right]-3}{15}\right)$, where [.] denotes greatest integer function, then the number of even integers for which $f\left(x\right)+g\left(x\right)$ is defined, is

Answer 1

If domain of $f\left(x\right)=D_1$ & Domain of $g\left(x\right)=D_2$
$\Rightarrow$ Domain of $f\left(x\right)+g\left(x\right)=D_1\cap D_2$
For $D_1$

$\frac{x-2}{x^2-10x+24}\ne 0\Rightarrow x\ne 2,4,6$
And for $D_2$

$-1\le \frac{2\left[x\right]-3}{15}\le 1\Rightarrow -6\le \left[x\right]\le 9$
$\Rightarrow x\in [-6,10)$
So the possible even integers are even integers in $D_1\cap D_2=\{-6,-4,-2,0,8\}$
Only $5$ integers, hence answer is $5$.