Question

Find the domain of the rational function.

$R\left(x\right)=\frac{x}{x^3-8}$

Short answer

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

Step-by-step solution

Step 1. Given information

The given function is$R\left(x\right)=\frac{x}{x^3-8}$

Step 2. Find the domain  

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

so,

$$\begin{gathered} x^3-8\ne 0 \\ {x}^3-2^3\ne 0 \\ x\ne 2 \end{gathered}$$

Thus, the domain of the given function is all real numbers except $2$.

Therefore, the domain is

$\hspace{0.17em}\left(-\infty \hspace{0.17em},\hspace{0.17em}2\right)\cup \left(2,\hspace{0.17em}\infty \hspace{0.17em}\right)$