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<y≤0[/latex]. It can be written in interval notation as [latex]y\in(-\infty,\,-1]\cup(0,\,+\infty)[/latex] or in set notation as [latex]\{y\;|\;y \in \mathbb{R}[/latex], \ [latex]-1<y≤0\}[/latex]. The symbol \ means except. There is a horizontal asymptote at [latex]y=0[/latex] (Figure 1: top right). There will always be a horizontal asymptote at [latex]y=0[/latex] when the degree of [latex]P(x)[/latex] is less than the degree of [latex]Q(x)[/latex]. In this case, the degree of [latex]P(x)[/latex] is 0, which is less than the degree of [latex]Q(x)[/latex], which is 2.

Example 1

State the domain and range of the function [latex]f(x)=\dfrac{x^2+3x-11}{x^2-4}[/latex].

Zoomed in view of [latex]f(x)=\dfrac{x^{2}+3x-11}{x^2-4}[/latex]

Solution

This function is graphed in figure 1 (bottom left). It has vertical asymptotes at [latex]x=\pm2[/latex], therefore [latex]x=\pm2[/latex] are restrictions on the domain.

In interval notation Domain = [latex](-\infty,\,-2)\cup(-2,\,2)\cup(2,\,+\infty)[/latex]

In set notation Domain = [latex]\{x\;|\;x \in \mathbb{R}, x\neq \pm2\}[/latex]

 

The graph has a horizontal asymptote at [latex]y=1[/latex] but this asymptote is crossed by the function just at [latex]x=2.333[/latex] so must be included in the range. There is however a gap in the range values from the highest point on the lower right part of the graph and the turning point on the upper part of the curve. We can use Desmos to determine the approximate value of these points by moving the curser over the graph. So, the range is [latex]y\in(-\infty,\,1.424]\cup[2.326,\,+\infty)[/latex] in interval notation, and [latex]\{y\;|\;y \in \mathbb{R}, x\neq 1\}[/latex] in set notation.

Vertical Asymptotes

Have you noticed the relationship between the vertical asymptotes and the restrictions on the domain?

For example,

A vertical asymptote is a vertical line that passes through the restricted value of the domain. The vertical asymptote can never be crossed by the graph, since the graph is undefined at that value.

We can easily find any vertical asymptotes by determining the restrictions on the domain. For a rational function [latex]f(x)=\dfrac{P(x)}{Q(x)}[/latex] the domain will be restricted when the denominator, [latex]Q(x)[/latex], is equal to zero. This is because division by zero is undefined.

Horizontal Asymptotes

Horizontal asymptotes appear on the graph of a rational function [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].