Question

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

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

Answer 1

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

Given: $f\left(x\right)=\sqrt{\frac{1-5^x}{7^{-x}-7}}$
For domain,
$\frac{1-5^x}{7^{-x}-7}\ge 0$
Case$-\left(i\right):$
$1-5^x\ge 0$ and $7^{-x}-7>0$
$\Rightarrow x\le 0\text{and}x<-1\Rightarrow x\in (-\infty ,-1)\cdots \left(i\right)$
Case $-\left(ii\right):$
$1-5^x\le 0$ and $7^{-x}-7<0$
$\Rightarrow x\ge 0\text{and}x>-1\Rightarrow x\in [0,\infty )\cdots \left(ii\right)$

Taking $\left(i\right)\cup \left(ii\right)$
Domain is
$x\in (-\infty ,-1)\cup [0,\infty )$