Question

Find the domain of definition of the following function.
$f\left(x\right)=\sqrt{x^2-x-20}+\sqrt{6-x}.$

Answer 1

$f\left(x\right)=\sqrt{x^2-x-20}+\sqrt{6-x}$

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

$(6-x)\ge 0$

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

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

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

$x\ge -4$

$x\ge 5$

$(6-x)\ge 0$

$-x\ge -6$

$x\le 6$

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

$x+4\le 0$ and $x-5\le 0$

or, $x\in (-\infty ,-4]$

So,the Domain of $f\left(x\right)=(-\infty ,-4]\cup [5,6]$