Question

Find the domain of definition of the following function:
$y=arc\cos \frac{2x+1}{2\sqrt{2x}}$

Answer 1

$y=\text{arc}\cos \left(\frac{2x+1}{2\sqrt{2x}}\right)$

$-1\le \frac{2x+1}{2\sqrt{2x}}\le 1$

$-2\sqrt{2x}\le 2x+1,$

$⟹8x\le 4x^2+4x+1,$

$⟹4x^2-4x+1\ge 0,$

$⟹{(2x-1)}^2\ge 0$

This verifies for all $x.$

${(2x-1)}^2\le {\left(2\sqrt{2x}\right)}^2$

$⟹4x^2+4x+1\le 8x,$

$⟹{(2x-1)}^2\le 0,$

$⟹2x-1=0,$

$⟹x=\frac{1}{2},$

$\therefore$ So, domain will be $\frac{1}{2}.$