Question

If f(x)=sin log $\left(\sqrt{4-x^2}/\left(1-x\right)\right)$ then the domain and range f are (respectively)

• A. $[-1,1],(-1,1)$
• B. $(-2,1),(-1,1)$
• C. $(1,2),[-1,1]$
• D. $(-2,1),[-1,1]$

Answer 1

The correct option is D $(-2,1),[-1,1]$

consider the given function ,

$f\left(x\right)=\sin {\log }_e\left(\frac{\sqrt{4-x^2}}{1-x}\right)$

Now ,

for ${\log }_e\frac{\sqrt{4x^2}}{1-x}$ to be defined $\frac{\sqrt{4-x^2}}{1-x}>0$

$\frac{\sqrt{4-x^2}}{1-x}>0$

$\sqrt{4-x^2}>0$, $x\ne 1$ and $1-x>0$

$4-x^2>0$ $1>x$

$4>x^2$ $x<1$

$x^2-4<0$

$(x+2)(x-2)<0$

$-2<x<2$

combining the two we get

$-2<x<1$

$xϵ(-2,1)$

Hence , the domain of the function is $(-2,1)$

Hence, this is the range of the function $[-1,1]$