# What is the domain of F(x)= (x-2)/( x^3+x)?

Nov 8, 2017

#### Answer:

Domain: $\left(- \infty , 0\right) \cup \left(0 , + \infty\right)$

#### Explanation:

$F \left(x\right) = \frac{x - 2}{{x}^{3} + x}$

$= \frac{x - 2}{x \left({x}^{2} + 1\right)}$

$F \left(x\right)$ is defined for all $x$ except where $x \left({x}^{2} + 1\right) = 0$

Since $\left({x}^{2} + 1\right) \ge 1 \forall x \in \mathbb{R}$

$\to F \left(x\right)$ is defined $\forall x \in \mathbb{R} : x \ne 0$

Hence the domain of $F \left(x\right)$ is $\left(- \infty , 0\right) \cup \left(0 , + \infty\right)$

As can be deduced from the graph of $F \left(x\right)$ below.

graph{(x-2)/(x^3+x) [-10, 10, -5, 5]}