Question

The domain of definition of the function $f\left(x\right)={\log }_{10}\left(\sqrt{10\cdot 3^{x-2}-9^{x-1}-1}\right)+\sqrt{{\cos }^{-1}(1-x)}$ is

• A. $[0,1]$
• B. $[1,2]$
• C. $(0,2)$
• D. $(0,1)$

Answer 1

The correct option is C $(0,2)$

For $f$ to be defined,
${\cos }^{-1}(1-x)\ge 0$
$\Rightarrow -1\le 1-x\le 1$
$\Rightarrow -2\le -x\le 0$
$\Rightarrow 0\le x\le 2\cdots \left(1\right)$

Also, $10\cdot 3^{x-2}-9^{x-1}-1>0$
$\Rightarrow 10\cdot 3^x-9^x-9>0$
$\Rightarrow 10\cdot 3^x-3^{2x}-9>0$
$\Rightarrow 3^{2x}-10\cdot 3^x+9<0$
$\Rightarrow (3^x-1)(3^x-9)<0$
$\Rightarrow 1<3^x<9$
$\Rightarrow 0<x<2\cdots \left(2\right)$
From $\left(1\right)$ and $\left(2\right),$ we have
$x\in (0,2)$