Question

Find the domain of the rational function.

$G\left(x\right)=\frac{6}{\left(x+3\right)\left(4-x\right)}$

Short answer

The domain of the given function is$\left(-\infty \hspace{0.17em},\hspace{0.17em}-3\right)\cup \left(-3,\hspace{0.17em}4\right)\cup \left(4,\hspace{0.17em}\infty \hspace{0.17em}\right)$

Step-by-step solution

Step 1. Given information   

The given function is$G\left(x\right)=\frac{6}{\left(x+3\right)\left(4-x\right)}$

Step 2. Find the domain   

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

So,

$$\begin{gathered} \left(x+3\right)\left(4-x\right)\ne 0 \\ x+3\ne 0 \\ x\ne -3 \\ or \\ 4-x\ne 0 \\ x\ne 4 \end{gathered}$$

Thus, the domain of the given function is all real numbers except $-3$and $4$.

Therefore, the domain is $\hspace{0.17em}\left(-\infty \hspace{0.17em},\hspace{0.17em}-3\right)\cup \left(-3,\hspace{0.17em}4\right)\cup \left(4,\hspace{0.17em}\infty \hspace{0.17em}\right)$