How do you find the domain for f(x) = (4x^2 - 9)/(x^2 + 5x + 6)?

Jun 7, 2015

To find the domain, we only need to factor the denominator to find its zeros. Other than the denominator being zero, $f \left(x\right)$ is well defined for all $x \in \mathbb{R}$.

${x}^{2} + 5 x + 6 = \left(x + 3\right) \left(x + 2\right)$

which is zero when $x = - 3$ or $x = - 2$

So the domain of $f \left(x\right)$ is:

$\left(- \infty , - 3\right) \cup \left(- 3 , - 2\right) \cup \left(- 2 , \infty\right)$

That is $\left\{x \in \mathbb{R} : x \ne - 3 \mathmr{and} x \ne - 2\right\}$