Question

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

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

Answer 1

Answer: (C)

We have for ${\cos }^{-1}(1-x)\ge 0\Rightarrow -1\le (1-x)\le 1\Rightarrow -2\le -x\le 0\Rightarrow 0\le x\le 2$ ....(1) also
$10\cdot 3^{x-2}-9^{x-1}-1>0$
$10\cdot 3^x-9^x-9>0$
$10\cdot 3^x-3^{2x}-9>0$
$3^{2x}-10\cdot 3^x+9<0$
$\left(3^x-1\right)\left(3^x-9\right)<0$
$1<3^x<9\Rightarrow 0<x<2$.....(2)
from (1) and (2)
$0<x<2$