Question

Let $D$ be the domain of the function $f\left(x\right)=\sin \left(\log \frac{\sqrt{9-x^2}}{1-x}\right).$ Then the number of integers in $D$ is

Answer 1

$f\left(x\right)=\sin \left(\log \frac{\sqrt{9-x^2}}{1-x}\right)$
Here, $9-x^2\ge 0$
$\Rightarrow x\in [-3,3]\cdots \left(1\right)$
and $\frac{\sqrt{9-x^2}}{1-x}>0$
$\Rightarrow 1-x>0,x\ne \pm 3$ $(\because \text{for}\log x,x\ne 0)$
$\Rightarrow x\in (-\infty ,-3)\cup (-3,1)\cdots \left(2\right)$
From $\left(1\right)$ and $\left(2\right),$
$D=(-3,1)$
Integers in $D$ are $-2,-1,0$