Question

Find the domain of the rational function.

$\frac{-4x^2}{\left(x-2\right)(x+4)}$

Short answer

The domain of the given function is$\hspace{0.17em}\left(-\infty ,\hspace{0.17em}-4\right)\cup \left(-4,\hspace{0.17em}2\right)\cup \left(2,\hspace{0.17em}\infty \hspace{0.17em}\right)$

Step-by-step solution

Step 1. Given information  

The given function is$H\left(x\right)=\frac{-4x^2}{\left(x-2\right)\left(x+4\right)}$

Step 2. Find the domain  

For the domain, the denominator should not be equal to zero.

So,

$$\begin{gathered} \left(x-2\right)(x+4)\ne 0 \\ x-2\ne 0 \\ x\ne 2 \\ or \\ x+4\ne 0 \\ x\ne -4 \end{gathered}$$

Thus, the domain of the given function is all real numbers except $2$and $-4$

Therefore, the domain islocalid="1646074172973" $\hspace{0.17em}\left(-\infty ,\hspace{0.17em}-4\right)\cup \left(-4,\hspace{0.17em}2\right)\cup \left(2,\hspace{0.17em}\infty \hspace{0.17em}\right)$