Question

Find the domain of the following functions:
$f\left(x\right)=\frac{x-3}{(x+3)\sqrt{x^2-4}}$

Answer 1

Given, $f\left(x\right)=\frac{x-3}{(x+3)\sqrt{x^2-4}}$

Equating the denominator to zero i.e.,

$(x+3)\sqrt{x^2-4}=0$

$\Rightarrow x+3=0$ or $\sqrt{x^2-4}=0$

$\Rightarrow x=-3$ or $x^2-4=0$

$\Rightarrow x^2=4$

$\Rightarrow x=\pm 2$

$\therefore$ Domain of $f\left(x\right)$ is all real numbers except $-3,\pm 2$

i.e., $D=(-\infty ,-3)\cup (-3,-2)\cup (-2,2)\cup (2,\infty )$.