Question

Graph the function g(x) = -3 / (x²- 4) . What is the domain of g(x)? Explain your reasoning. Do not completely trust calculator or computer graphs of this function.

Answer 1

$\bf g(x)=-\cfrac{3}{x^2-4}\\\\ ------------------$

if the value of the denominator is 0
the fraction becomes "undefined"
so, any value of "x" that makes it 0
is not a valid value for "x", and thus
not part of the domain
let's set the denominator to 0, and find out which one(s) are those if any

$\bf x^2-4=0\implies x^2=4\implies x=\pm\sqrt{4}\implies x=\pm 2\\\\ ------------------\\\\ so \\\\ g(x)=-\cfrac{3}{(-2)^2-4}\to -\cfrac{3}{4-4}\implies g(x)=-\cfrac{3}{0} \\\\\\ g(x)=-\cfrac{3}{(2)^2-4}\implies g(x)=-\cfrac{3}{4-4}\implies g(x)=-\cfrac{3}{0}$

$\bf -----------------------------\\\\ thus\qquad domain\implies \{x|x\in \mathbb{R};x\ne \pm 2\} \\\\ \textit{or in interval notation}\implies (-\infty,-2)\cup(2,+\infty)$