Question

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

Answer 1

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

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

$x^2-5x-14>0$

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

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

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

$x=-4,5$

$x^2-5x-14>0$

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

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

$x=-2,7$

So, the domain of $y=(-\infty ,-4)\bigcup [7,\infty )$