Question

The domain of $f\left(x\right)=\sqrt{x^2-x-20}+\frac{1}{\sqrt{x^2-5x-14}}$ is

• A. $(-\infty ,-2)\cup (7,\infty )$
• B. $(-\infty ,-4]\cup (7,\infty )$
• C. $(-\infty ,-2)\cup [5,\infty )$
• D. $[-4,7)$

Answer 1

The correct option is B $(-\infty ,-4]\cup (7,\infty )$

$f\left(x\right)=\sqrt{x^2-x-20}+\frac{1}{\sqrt{x^2-5x-14}}$
For $\sqrt{x^2-x-20}$ to be defined
$\Rightarrow x^2-x-20\ge 0$
$\Rightarrow (x+4)(x-5)\ge 0$
$\Rightarrow x\le -4\text{or}x\ge 5...\left(1\right)$

For $\frac{1}{\sqrt{x^2-5x-14}}$ to be defined
$\Rightarrow x^2-5x-14>0$
$\Rightarrow (x+2)(x-7)>0$
$\Rightarrow x<-2\text{or}x>7...\left(2\right)$

Domain of $f\left(x\right)=\left(1\right)\cap \left(2\right)$
$=(-\infty ,-4]\cup (7,\infty )$