Question

# Determine the domains of the given rational functions. (x^2-9)/(x^3-x)

Rational functions
Determine the domains of the given rational functions.
$$\displaystyle\frac{{{x}^{{2}}-{9}}}{{{x}^{{3}}-{x}}}$$

2021-06-21

By rules of the domain ofthe function that denominator can not be zero, we have
$$\displaystyle{x}^{{3}}-{x}\frac{=}{{0}}$$
$$\displaystyle{x}^{{3}}-{x}$$ can be written as $$\displaystyle{x}^{{3}}-{x}={x}{\left({x}^{{2}}-{1}\right)}$$
From here we have that
$$\displaystyle{x}^{{3}}—{x}={x}{\left({x}^{{2}}-{1}\right)}={0}$$ when $$x = 0$$ or $$\displaystyle{x}^{{2}}—{1}={0}$$.
$$\displaystyle{x}^{{2}}—{1}={0}$$ is quadratic function and we have
$$\displaystyle{x}^{{2}}={1}$$
Then domain of the given function is
$$R\{-1,0,1\}$$