2.5 Rational Functions

In this section, we will discuss the basics of rational functions. A rational function is a function that can be written as the quotient of two polynomial functions where the function in the denominator is nonzero. Formally, suppose there are two polynomial functions $P(x)$ and $Q(x)$ such that $Q(x)$ is a nonzero polynomial. A function of the following form is called a rational function:

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

The following are some examples of rational functions:

$\begin{array}{rl}& f(x)=\frac{3}{x+5}\\ & f(x)=\frac{x^2-3x+1}{x-4}\\ & f(x)=\frac{5x^7+4x^3-2x^2+10x+3}{2}\end{array}$

The domain of a polynomial is the set of all real numbers, while the domain of a rational function is the set of all real numbers that do not make the denominator of the function equal to 0. For example, the domain of

$f(x)=\frac{3}{x+5}$

is $x<-5$ and $-5<x$, because -5 makes the denominator equal to 0.

What is the domain of the following rational function?

$f(x)=\frac{x-3}{x^2-5x+4}$

To find the domain, we set the denominator equal to 0, and then solve for $x$.

$\begin{array}{rl}{x}^2-5x+4& =0\\ (x-4)(x-1)& =0\\ x& =1,4\end{array}$

Therefore, the domain of this function is $x<1$, $1<x<4$, and $4<x$.

Suppose a rational function $f(x)$ is

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

The function $f(x)$ is positive if $P(x)>0$ and $Q(x)>0$, or if $P(x)<0$ and $Q(x)<0$. On the other hand, $f(x)$ is negative if $P(x)<0$ and $Q(x)>0$ , or if $P(x)>0$ and $Q(x)<0$. In other words, if the numerator and denominator have the same signs, then the rational function is positive. If the numerator and denominator have opposite signs, then the rational function is negative.

Let’s look at another function:

$f(x)=\frac{x-3}{x^2-5x+4}$

The numerator, $P(x)$, is positive if $x$ is greater than 3 and negative if $x$ is less than 3. The denominator, $Q(x)$ is positive if $x$ is less than 1 or greater than 4 and negative if $x$ is greater than 1 and less than 4.

Therefore, $f(x)$ is positive when $1<x<3$ or when $4<x$ , and negative when $x<1$ or when $3<x<4$. The function $f(x)$ is 0 when $x=3$ since the numerator is 0, and $f(x)$ cannot be defined when $x=1$ or $x=3$ since the denominator equals 0.

While there are a few important topics regarding rational functions that would be discussed in a full-length math class, such as asymptotes, graphs, and holes, we will not discuss them in this text since our purpose is to give you a quick refresher to prepare you for a graduate program.