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Section 9.3 Rational Functions

Objectives: Student Learning Outcomes

After completing this activity you will be able to:

  1. Determine the domain, range, intercepts, holes, end behavior, vertical and horizontal asymptotes (if they exist) of a rationalfunction.

  2. Create a graph by hand and on the calculator of a rational function (given in factored form).

  3. Maintain and Strengthen Prerequisites.

Subsection 9.3.1 Rational Function Definition

Basically, a rational funciton is a ratio (fraction). It is composed of a numerator and a denominator in which both are polynomials. If a function can be written as

\begin{equation*} f(x)=\frac{\text{polynomial}}{\text{polynomial}} \end{equation*}

then it is a rational function.

The functions listed below are rational functions.

  • \(\frac{x^{2}-5x}{10-3x^{3}+22x^{7}}\)

  • \(\frac{4}{x}\)

  • \(\frac{100x^{10}-x}{x+1}\)

  • \(\frac{1}{x}+\frac{x^{2}}{x-5}\text{.}\) In its current form this expression is the sum of two rational functions. Later we will see how we can combine the terms to write the expression as a ratio of polynomials.

Notice the definition states specifically that the ratio is made up of polynomials. That is important to remember. The functions listed below are NOT rational funcitons because either the numerator or the denominator is not a polynomial.

  • \(\frac{\sqrt{x}+1}{3x^{4}-x}\text{.}\) The square root in the numerator is the equivalent of \(x^{\frac{1}{2}}\) which is not a polynomial.

  • \(\frac{2x-x^{7}}{x^{4}-6^{x}}\text{.}\) In the denominator the term \(6^{x}\) is exponential not a polynomial.

You may already be wondering if power functions have anything in common with polynomials and rational functions. The answer is yes, sometimes.

Some power functions are rational, some power functions are polynomials, and some power functions are neither rational nor polynomials.

  1. \(\frac{1}{x^{2}}\) is a power function because it can be written as \(x^{-2}\text{,}\) but it is also a rational function because the numerator is \(1\text{,}\) a polynomial of degree zero, and the denominator \(x^{2}\) is a polynomial of degree two.

  2. Any power function with non-zero, integer expoent is a polynomial. Consider

    \begin{gather*} x^{2}\\ x^{10}\\ x^{315} \end{gather*}
  3. Consider the power function \(x^{\frac{-1}{2}}\text{.}\) It is not a polynomial because the exponent is not a non-negative integer and it is not a rational function because it cannot be written as a ratio of polynomials.

Subsection 9.3.2 Applications of Rational Functions

Rational functions sometimes occur when making comparisons between two quantities, hence the ratio.

Consider how the percent copper in a metal alloy changes as we add grams of copper to 50 grams of alloy. Intially the alloy has no copper in it.

Find a formula to model the concentration of coppoer as a function of the amount of copper added to the alloy mixture.

Sometimes a function will not look rational but can be converted into a ratio of polynomials.

A can of food is constructed of two different materials, a light material for the circular top and bottom of the can and a sturdier, thick material for the side of the can. The two materials have different costs so the total cost of making the can is the sum of costs of its pieces. Let

\begin{equation*} C(r)=\frac{0.6}{r}+0.2r^{2} \end{equation*}

be the cost of making a can of radius r.

\(C(r)\) is the sum of two rational functions, so we can combine the two terms in the equation by creating a common denominator between the two them.

\begin{align*} C(r) \amp =\frac{0.6}{r}+0.2r^{2}\\ \amp=\frac{0.6}{r}+0.2r^{2} \cdot \frac{r}{r}\\ \amp=\frac{0.6}{r}+\frac{0.2r^{3}}{r}\\ \amp=\frac{0.6+0.2r^{3}}{r} \end{align*}

After combining the terms we arrive at an expression that is now a ratio of two polynomials.

Subsection 9.3.3 Long Run Behavior

Rational Functions have distinctive behaviors for inputs far from the origin versus inputs close to the origin. When consider the characteristics of a funciton when the inputs are far from the origin, we call it long-run behavior.

The graph initially shows two functions from a “far away” perspective. In math we call this the long-run behavior.

Subsection 9.3.4 Short-run Behavior

Short-run behavior refers to any characteristics or special traits a function may have when the inputs are “close” to the origin.

In math “close” is any value that can be placed on a number line. In fact, if a value is NOT infinity, then it is considered close. As an application, three billion miles may seem far away, but it's nothing when compared to the size of the universe!

In the following execises you will explore the behavior of some rational functions for values near the origin.

Subsection 9.3.5 Building a Rational Function

Using similar techniques for building a polynomial, we can use the zeros, vertical and horizontal asymptotes as guides to help us construct possible formulas for rational functions.

In the next exercise you will use specific properties of the graph of a rational function to construct a possible formula for a rational function.