Question

The domain of definition of the function $f\left(x\right)=\frac{\log \left(2x-3\right)}{\sqrt{x-1}}+\sqrt{5-2x}$ is

• A. $\left[1,\frac{5}{2}\right)$
• B. $\left[\frac{3}{2},\frac{5}{2}\right)$
• C. $\left[1,\frac{5}{2}\right]$
• D. $\left(\frac{3}{2},\frac{5}{2}\right]$

Answer 1

Answer: (D)

$f\left(x\right)=\frac{\log \left(2x-3\right)}{\sqrt{x-1}}+\sqrt{5-2x}$
$\sqrt{x-1}>0\Rightarrow x>1$ and $\log \left(2x-3\right)>0$
$\Rightarrow 2x-3>0\Rightarrow x>\frac{3}{2}$
and $\sqrt{5-2x}\ge 0\Rightarrow 5\ge 2x\Rightarrow x\le \frac{5}{2}$
$\therefore \frac{3}{2}<x\le \frac{5}{2}$
Hence, domain of $f\left(x\right)$ is $\left(\frac{3}{2},\frac{5}{2}\right]$