Question

The domain of the following function $f\left(x\right)=\sqrt{\frac{1-5^x}{7^{-x}-7}}$ is

• A. $(-\infty ,-1)\cup (0,\infty )$
• B. $(-\infty ,-1)\cup [0,\infty )$
• C. $(-\infty ,1)\cup [7,\infty )$
• D. $(-\infty ,-7)\cup [7,\infty )$

Answer 1

The correct option is B $(-\infty ,-1)\cup [0,\infty )$

$f\left(x\right)=\sqrt{\frac{1-5^x}{7^{-x}-7}}$
For $f\left(x\right)$ to be defined (i) $1-5^x\ge 0\Rightarrow 1\ge 5^x\Rightarrow x\le 0$
And $7^{-x}-7>0$
$\Rightarrow x<-1$
$\Rightarrow x\in (-\infty ,-1)$
or (ii) $1-5^x\le 0\Rightarrow x\ge 0$
& $7^{-x}-7<0\Rightarrow x>-1$
$\Rightarrow x\in [0,\infty )$
Thus domain is $x\in (\infty ,-1)\cup [0,\infty )$