Question

The domain of the definition of the function $f\left(x\right)={\sin }^{-1}\left(\frac{|x|-5}{2}\right)+{\left[{\log }_{10}\left(6-x\right)\right]}^{-1}$ is :

• A. $\left(7,7\right)$
• B. $\left(-7,-1)\cup (-3,7\right)$
• C. $\left[-7,-3\right]\cup [3,5)\cup \left(5,6\right)$
• D. $(-3,4)\cup \left(5,6\right)$

Answer 1

The correct option is C $\left[-7,-3\right]\cup [3,5)\cup \left(5,6\right)$

$f\left(x\right)={\sin }^{-1}\left(\frac{\left|x\right|-5}{2}\right)+{\left[{\log }_{10}(6-x)\right]}^{-1}-1\le \frac{|x-5|}{2}\le 1-2\le \left|x\right|-5\le 23\le \left|x\right|\le 76-x>0x<66-x\ne 1x\ne 53\le \left|x\right|(x-3)(x+3)\ge 0\left|x\right|\le 7x\le \pm 7(x+7)(x-7)\le 0xϵ[-7,-3]\cup [3,5)\cup (5,6)$