Rational Functions

A rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials, nor the values taken by the function, are necessarily rational numbers.

Any function of one variable, $x$, is called a rational function if, and only if, it can be written in the form:

$f(x) = \dfrac{P(x)}{Q(x)}$

where $P$ and $Q$ are polynomial functions of $x$ and $Q(x) \neq 0$. 

Note that every polynomial function is a rational function with $Q(x) = 1$. A function that cannot be written in the form of a polynomial, such as $f(x) = \sin(x)$, is not a rational function. However, the adjective "irrational" is not generally used for functions.

A constant function such as $f(x) = \pi$ is a rational function since constants are polynomials. Note that the function itself is rational, even though the value of $f(x)$ is irrational for all $x$. 

The Domain of a Rational Function

The domain of a rational function $f(x) = \frac{P(x)}{Q(x)}$ is the set of all values of $x$ for which the denominator $Q(x)$ is not zero. 

For a simple example, consider the rational function $y = \frac {1}{x}$. The domain is comprised of all values of $x \neq 0$.

Domain restrictions can be calculated by finding singularities, which are the $x$-values for which the denominator $Q(x)$ is zero. The rational function is not defined for such $x$-values, and these values are excluded from the domain set of the function. 

Factorizing the numerator and denominator of rational function helps to identify singularities of algebraic rational functions. Singularity occurs when the denominator of a rational function equals $0$, whether or not the linear factor in the denominator cancels out with a linear factor in the numerator. 

Example 1

Consider the rational function

 $f(x) = \dfrac{(x^2 - 3x -2)}{(x^2 - 4)}$

The domain of this function includes all values of $x$, except where $x^2 - 4 = 0$. 

We can factor the denominator to find the singularities of the function: 

$x^2 - 4 = (x + 2)(x - 2)$

Setting each linear factor equal to zero, we have $x+2 = 0$ and $x-2 = 0$. Solving each of these yields solutions $x = -2$ and $x = 2$; thus, the domain includes all $x$ not equal to $2$ or $-2$. This can be seen in the graph below.

The domain of a function

Graph of a rational function with equation $\frac{(x^2 - 3x -2)}{(x^2 - 4)}$. The domain of this function is all values of $x$ except $+2$ or $-2$.

Note that there are vertical asymptotes at $x$-values of $2$ and $-2$. This means that, although the function approaches these points, it is not defined at them.

Example 2

Consider the rational function

 $f(x)= \dfrac{(x + 3)}{(x^2 + 2)}$

The domain of this function is all values of $x$ except those where $x^2 + 2 = 0$. However, for $x^2 + 2=0$ , $x^2$ would need to equal $-2$. Since this condition cannot be satisfied by a real number, the domain of the function is all real numbers.