Rational Functions: Concepts and Real-World Applications

By Jeff Twiddy

Rational functions are a cornerstone of precalculus, appearing in various real-life scenarios. In essence, a rational function is any function that can be written as the quotient of two polynomials. That is, if

where both P(x) and Q(x) are polynomials and Q(x) ≠0, thenf(x) is a rational function.

This article will not only review the core concepts of rational functions but also illustrate their practical applications through real-world examples that resonate with your students. After all, when trying to motivate students in your math class, it’s helpful if they can see how it applies in the world they live in – it’s not just a thing they need to know to pass a test. Rather, emerging from the misconception that an equation does not pertain to anything in real life, they become empowered to think deeply and problem-solve for the things that are, in fact, real-life and part of their life.

Key Concepts and Properties of Rational Functions

Domain and Discontinuities

The domain of a rational function includes all real numbers except where the denominator Q (x) is zero. These values often create holes or vertical asymptotes in the graph.

Vertical asymptotes occur when Q(x) =0 (and the corresponding factor doesn’t cancel with the numerator).
Holes arise when a factor in Q(x) cancels with a factor in P(x), indicating a removable discontinuity.

Asymptotes

Rational functions frequently exhibit horizontal or slant asymptotes:

Horizontal asymptotes describe the end behavior when the degrees of P(x) and Q(x) are equal or when the degree of P(x) is less than that of Q(x).
A slant (oblique) asymptote may occur if the degree of P(x) is exactly one more than the degree of Q(x). In such cases, polynomial long division can reveal the slant asymptote.

Graph Behavior and Intercepts

Understanding intercepts is crucial:

x-intercepts are found by setting P(x) =0 (provided those zeros are not canceled by Q(x).
y-intercepts can be computed by evaluating f(0), as long as x = 0 is within the domain.

Graphing rational functions often involves plotting these key features – domain restrictions, intercepts, and asymptotes – to understand the overall behavior of the function.

Real-World Applications of Rational Functions

Rational functions are more than abstract equations; they serve as powerful models in many practical situations. Below are several examples that high school students can relate to:

Economics: Average Cost Functions

In business, average cost functions are commonly modeled by rational functions. Consider a production scenario where a company has a fixed cost F (such as rent or equipment) and a variable cost per unit v. The total cost for producing x units is given by:

Total Cost = F + v ⋅ x

The average cost per unit then becomes:

This rational function shows that as production x increases, the average cost approaches the variable cost v, highlighting the benefits of economies of scale.

Physics and Engineering: Resistance in Parallel Circuits

In electrical engineering, the total resistance R_T of resistors connected in parallel is determined by a rational function. For two resistors R₁ and R₂, the total resistance is:

This formula is a rational function where the resistances in the numerator and denominator determine how the overall circuit behaves.

Chemistry: Concentration in Mixture Problems

Consider a situation in chemistry where you mix two solutions with different concentrations. If you have a tank with a mixture and you’re adding a solution at a certain rate while simultaneously draining the mixture, the concentration of the chemical can be modeled by a rational function. For example, if the concentration C(t) changes over time t due to inflow and outflow, the rate of change can often be expressed as:

where a, b, c, and d are constants related to the rates of mixing, addition, and removal. Such models help predict when the concentration will reach a desired level.

Medicine: Dosage and Drug Concentration

Rational functions can model how the concentration of a drug in the bloodstream changes over time. If the body metabolizes a drug at a rate proportional to its current concentration, the concentration C(t) might be modeled as:

Here, D is the initial dosage, and k is a constant related to the rate of metabolism. This function helps medical professionals understand how long it takes for the drug concentration to drop to a safe level.

Graphing Rational Functions Step-by-Step

Identify the Domain

Determine where the denominator Q(x) is zero. Mark these points on your graph as vertical asymptotes or holes.

Calculate Intercepts

x-intercepts: Solve P(x) = 0.
y-intercepts: Evaluate f(0) if 0 is in the domain.

Find Asymptotes

Vertical asymptotes: Set Q(x) = 0 (after canceling common factors).
Horizontal/slant asymptotes: Compare the degrees of P(x) and Q(x) or perform long division.

Sketch the Graph

Using the information above, draw the asymptotes and plot key points. Then, sketch the curve, noting how it behaves near discontinuities and as x approaches infinity.

Conclusion

Rational functions provide an accessible yet rich field of study in precalculus. By understanding their structure, properties, and behavior, students can see how these functions model real-life phenomena—from economics and engineering to chemistry and medicine. Exploring these practical examples not only reinforces the abstract concepts but also demonstrates the relevance of mathematics in everyday decision-making and problem-solving.

By mastering rational functions, you build a strong foundation for advanced studies in mathematics, science, and engineering—showing that even seemingly theoretical concepts have tangible real-world applications.