Definition of Rational Functions

A rational function has the form [latex]f(x)=\dfrac{P(x)}{Q(x)}[/latex], where [latex]P(x)[/latex] and [latex]Q(x)[/latex] are polynomials. In addition, [latex]Q(x)\neq 0[/latex]. The expression [latex]\dfrac{P(x)}{Q(x)}[/latex] is called a rational expression. Examples of rational functions include: [latex]f(x)=\dfrac{1}{x}[/latex],  [latex]f(x)=\dfrac{5x-3}{x^2-1}[/latex], [latex]h(x)=\dfrac{x^2-5}{x^3+2x^2+7}[/latex], [latex]g(x)=\dfrac{7x^3-5x}{x^2-5}[/latex].

Graphs of Rational Functions

We can use Desmos to graph rational functions. Try it out for yourself. Figure 1 shows some examples.

Figure 1. Examples of the graphs of rational functions

Domain and Range

Each function has a domain and a range. The domain is the set of all [latex]x[/latex]-values, while the range is the set of all [latex]y[/latex]-values.

The domain for the function [latex]f(x)=\dfrac{x-1}{x+4}[/latex] is all real numbers except [latex]x=-4[/latex]. It can be written in interval notation as [latex]x\in(-\infty,\,-4)\cup(-4,\,+\infty)[/latex] or in set notation as [latex]\{x\;|\;x \in \mathbb{R}, x\neq -4\}[/latex]. The point [latex]x=-4[/latex] is excluded from the domain because at that point the function value is undefined: [latex]f(-4)=\dfrac{-4-1}{-4+4}=\dfrac{-5}{0}[/latex], and we can’t divide by zero. [latex]x=-4[/latex] is called a restriction on the domain. The line [latex]x=-4[/latex] is a vertical asymptote (Figure 1: top left).

The range of the function [latex]f(x)=\dfrac{x-1}{x+4}[/latex] is all real numbers except [latex]y=1[/latex]. It can be written in interval notation as [latex]y\in(-\infty,\,1)\cup(1,\,+\infty)[/latex] or in set notation as [latex]\{y\;|\;y \in \mathbb{R}, y\neq 1\}[/latex]. There is a horizontal asymptote at [latex]y=1[/latex] (Figure 1: top left). Horizontal asymptotes appear on the graphs of rational functions [latex]f(x)=\dfrac{P(x)}{Q(x)}[/latex] when the degree of [latex]P(x)[/latex] is less than or equal to the degree of [latex]Q(x)[/latex]. In this case the degrees are equal at 1. When the degrees are equal, the horizontal asymptote is found by dividing the leading coefficients of [latex]P(x)[/latex] and [latex]Q(x)[/latex]. In this case, the horizontal asymptote is [latex]y=\dfrac{1}{1}=1[/latex].

The domain for the function [latex]f(x)=\dfrac{1}{x^2-1}[/latex] is all real numbers except [latex]x=-1[/latex] and [latex]x=1[/latex]. It can be written in interval notation as [latex]x\in(-\infty,\,-1)\cup(-1,\,1)\cup(1,\,+\infty)[/latex] or in set notation as [latex]\{x\;|\;x \in \mathbb{R}, x\neq \pm1\}[/latex]. The points [latex]x=-1[/latex] and [latex]x=1[/latex] are excluded from the domain because at those points the function value is undefined. [latex]x=-1[/latex] and [latex]x=1[/latex] are restrictions on the domain. The lines [latex]x=-1[/latex] and [latex]x=1[/latex] are vertical asymptotes (Figure 1: top right).

The range for the function [latex]f(x)=\dfrac{1}{x^2-1}[/latex] can be seen from the graph to exclude the [latex]y[/latex]-values greater than –1 and less than or equal to 0. That means the domain is all real numbers except [latex]-1

RESTRICTIONS ON THE DOMAIN

The rational function [latex]f(x)=\dfrac{P(x)}{Q(x)}[/latex] has restrictions on the domain when [latex]Q(x)=0[/latex].