Question

If range of the function $y=\sin \left[\log \left\{\frac{\sqrt{4-x^2}}{1-x}\right\}\right]$
is $[-1,a]$, what is $a$

Answer 1

${\log }_{a}z$ is defined if $z>0,z\ne 0$ Hence $\frac{\sqrt{4-x^2}}{1-x}>0,\ne 0$

$4-x^2=+ive$ or $x^2-4<0,(x+2)(x-2)<0$

$\therefore -2<x<2$ ...(1)

Now $\sqrt{4-x^2}$

is certainly +ive so we must have $1-x>0$ or $x<1$. ...(2) $\therefore$ Domain $=x:-2<x<1$

from (1) and (2)Range is clearly $[-1,1]$ as $-1\le \sin \theta \le 1$