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  4. Domain of definition of the function f(x)= log (√10 ⋅ 3x-2-9x-1-1)+√ cos -1(1-x) is

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Q. 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

Relations and Functions - Part 2

Solution:

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