Question

What are rational functions?

Answer 1

Definition of a Rational function.

A rational function is the ratio of two polynomial functions where the denominator polynomial is a non-zero polynomial.

A rational function $\mathrm{R}\left(x\right)$ is of the form $\mathrm{R}\left(x\right)=\frac{\mathrm{P}\left(x\right)}{\mathrm{Q}\left(x\right)}$ Where, $\mathrm{P}\left(x\right)\mathrm{and}\mathrm{Q}\left(x\right)$ are the polynomial of the type $a{x}^n+b{x}^{n-1}+...\alpha x+\beta$ and $x$ belongs to some interval of real number such that $Q\left(x\right)\ne 0$

The examples of rational functions are:

1. $R\left(x\right)=\frac{2x+5}{x-1},x\ne 1$

Here for $x=1$ denominator $\mathrm{Q}\left(x\right)=x-1=0$ thus the domain of $x$ such that $Q\left(x\right)\ne 0$ is $x\in \mathrm{ℝ}-\left\{1\right\}$

2. $R\left(x\right)=\frac{x^2+5x-6}{x^2+2},x\in R$

Here for$Q\left(x\right)\ne 0$ ,all $x\in \mathrm{ℝ}$, thus the domain of $x$ such that $Q\left(x\right)\ne 0$ is $x\in \left(-\infty ,\infty \right)$