Question

The domain of the function $f\left(x\right)={\log }_{1/2}(x-\frac{1}{2})+{\log }_2\sqrt{4x^2-4x+5}$ is

• A. $[\frac{1}{2},\infty )$
• B. $(\frac{1}{2},\infty )$
• C. $(-\infty ,\infty )$
• D. $(-\frac{1}{2},\infty )$

Answer 1

The correct option is B $(\frac{1}{2},\infty )$

$f\left(x\right)={\log }_{1/2}(x-\frac{1}{2})+{\log }_2\sqrt{4x^2-4x+5}$
${\log }_{1/2}(x-\frac{1}{2})$ is defined if $x-\frac{1}{2}>0$ i.e., $x>\frac{1}{2},$
and ${\log }_2\sqrt{4x^2-4x+5}$ is defined if
$4x^2-4x+5>0$
$\Rightarrow 4[{(x-\frac{1}{2})}^2+1]>0$ which is true for all real $x.$
$\therefore$ Domain of $f$ is $(\frac{1}{2},\infty )$