Question

Find the domain of the rational function.

$R\left(x\right)=\frac{x}{x^4-1}$

Short answer

The domain of the given function is$\hspace{0.17em}\left(-\infty \hspace{0.17em},\hspace{0.17em}-1\right)\cup \left(-1,\hspace{0.17em}1\right)\cup \left(1,\hspace{0.17em}\infty \hspace{0.17em}\right)$

Step-by-step solution

Step 1. Given information

The given function is$R\left(x\right)=\frac{x}{x^4-1}$

Step 2. Find the domain 

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

so,

$$\begin{gathered} x^4-1\ne 0 \\ {\left(x^2\right)}^2-1^2\ne 0 \\ \left(x^2-1\right)\left(x^2+1\right)\ne 0 \end{gathered}$$

We neglect $x^2+1$because the solution to this will be an imaginary number.

$$\begin{gathered} x^2-1\ne 0 \\ {x}^2\ne 1 \\ x\ne 1 \\ or \\ x\ne -1 \end{gathered}$$

Thus, the domain of the given function is all real numbers except $-1$and $1$.

Therefore, the domain isrole="math" localid="1646113099270" $\hspace{0.17em}\left(-\infty \hspace{0.17em},\hspace{0.17em}-1\right)\cup \left(-1,\hspace{0.17em}1\right)\cup \left(1,\hspace{0.17em}\infty \hspace{0.17em}\right)\hspace{0.17em}$