Question

Let $f:(-\infty ,+1]\to R,g:[-1,\infty )\to R$ be such that $f\left(x\right)=\sqrt{1-x}$ and $g\left(x\right)=\sqrt{1+x}$, then $f\left(x\right)+\frac{1}{g\left(x\right)}$ exist if $x\in$

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

Answer 1

The correct option is D $(-1,1]$

$f\left(x\right)+\frac{1}{g\left(x\right)}=\sqrt{1-x}+\frac{1}{\sqrt{1+x}}$

Now, $1-x\ge 0,1+x>0$
$\Rightarrow x\le 1,x>-1\therefore x\in (-1,1]$