Question

Find the domain of the following function.
$y=\frac{1}{\sqrt{14+5x-x^2}}+\sqrt{x^2-x-20}.$

Answer 1

$y=\frac{1}{\sqrt{{14+5x-x}^2}}+\sqrt{x^2-x-20}$

${-x}^2+5x+14>0$

$(x^2-x-20)>0$

${-x}^2+7x-2x+14>0$

$(x+2)(-x+7)>0$

$x>-2$

$x<7$

$(x^2-x-20)\ge 0$

$(x^2-5x+4x-20)\ge 0$

$x(x-5)+4(x-5)\ge 0$

$(x+4)(x-5)\ge 0$

$x\ge -4$

$x\ge 5$

So, the domain of $y=[5,7)$