Question

Find the domain of the following function.
$y=\sqrt{5-x-\frac{6}{x}}.$

Answer 1

$y=\sqrt{5-x-\frac{6}{x}}$

$(5-x-\frac{6}{x})\ge 0$

$\left(\frac{5x-x^2-6}{x}\right)\ge 0$

$x\ne 0$

$-x^2+5x-6\ge 0$

$x^2-5x+6\le 0$

$x^2-3x-2x+6\le 0$

$x(x-3)-2(x-3)\le 0$

$x=2,x=3$

When, $x>3$

$-x^2+5x-6<0$

When, $x<2$

$-x^2+5x-6<0$

So, the domain of $y=[2,3]$