# Rational Functions?

Jun 25, 2018

Yes, it is a rational function. $p \left(x\right) = {x}^{3} + 9$ and $q \left(x\right) = 9 x$.

#### Explanation:

$f \left(x\right) = {x}^{2} / 9 + \frac{1}{x}$
$= {x}^{2} / 9 \cdot 1 + \frac{1}{x} \cdot 1$
$= {x}^{2} / 9 \cdot \frac{x}{x} + \frac{1}{x} \cdot \frac{9}{9}$
$= {x}^{3} / \left(9 x\right) + \frac{9}{9 x}$
$= \frac{{x}^{3} + 9}{9 x}$

Let ${x}^{3} + 9 = p \left(x\right)$ and $9 x = q \left(x\right)$
$\therefore f \left(x\right) = \frac{p \left(x\right)}{q \left(x\right)}$

$f \left(x\right)$ can be expressed in the form of a rational fraction therefore it is a rational function. $p \left(x\right)$ matches the condition of having a leading coefficient of $1$. $p \left(x\right)$ and $q \left(x\right)$ share no common factor.