Question

Find the domain of the expression.

$\frac{x^2+1}{x^2-x-2}$

Short answer

The domain of the expression $\frac{x^2+1}{x^2-x-2}$is$D=\left\{x|x<-1or-1<x<2orx>2\right\}$

Step-by-step solution

Step 1. Apply the concept of domain.

The domain of an algebraic expression is the set of real numbers that the variable is permitted to have.

Step 2. Analyze the expression.

Given$\frac{x^2+1}{x^2-x-2}$

This radical is defined for every x such that

$x^2-x-2\ne 0$

Step 3. Find the domain.

Since the polynomial$\frac{x^2+1}{x^2-x-2}$ is defined for every x except

$\begin{array}{l}{x}^2-x-2=0\\ \Rightarrow x^2-2x+x-2=0\\ \Rightarrow x\left(x-2\right)+\left(x-2\right)=0\\ \Rightarrow \left(x-2\right)\left(x+1\right)=0\\ \Rightarrow x=2orx=-1\end{array}$

$D=\left\{x|x<-1or-1<x<2orx>2\right\}$