Question:
Mark (√) against the correct answer in the following:
Let $f(x)=\sqrt{\log }\left(2 x-x^{2}\right) .$ Then, dom $(f)=?$
A. $(0,2)$
B. $[1,2]$
C. $(-\infty, 1]$
D. None of these
Solution:
$f(x)=\sqrt{\log }\left(2 x-x^{2}\right)$
$2 x-x^{2}>1$
$\Rightarrow x^{2}-2 x+1<0$
$\Rightarrow(x-1)^{2}<0$
$\Rightarrow x-1<0$
$\Rightarrow x<1$
$\log \left(2 x-x^{2}\right)>0$
$\Rightarrow 2 x-x^{2}>e^{0}=1$
$\Rightarrow x<1$
$\operatorname{Dom}(f)=(-\infty, 1)$