Analyzing the Graph of a Rational Function: Asymptotes, Domain, and Range

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  • 0:03 Rational Functions
  • 0:49 Domain and Range
  • 3:47 Slant Asymptotes
  • 4:54 Lesson Summary
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Lesson Transcript
Instructor: Bob Bruner

Bob is a software professional with 24 years in the industry. He has a bachelor's degree in Geology, and also has extensive experience in the Oil and Gas industry.

A rational function arises from the ratio of two polynomial expressions. The graphs of rational functions often have distinct characteristics. In this lesson, we look at how to analyze some of those characteristics.

Rational Functions

Rational functions are defined as the ratio of two polynomial expressions. For example, suppose we're given two simple linear polynomial functions:

f1 = 10x + 6

f2 = x - 1

We can compose a rational function by simply taking their ratio.

f1 / f2 = (10x + 6) / (x - 1)

Graph 1

The graph of the resulting function is surprisingly complex for such simple inputs. We can see that this derived function consists of two distinct parts. Depending on the polynomial expressions found in the numerator and denominator, graphs of rational functions may take on various complex shapes. These shapes are defined in part by the effective domain and range of the function.

Domain and Range

The domain of a function consists of all the allowable values for the independent variable x. Similarly, the range of the function consists of all the possible values for the dependent variable y. If we look more closely at our graph, we see that it approaches, but never actually arrives at, certain values in x and y. In this case, both portions of the graph function are asymptotic to those values. Any asymptote that crosses the x-axis parallel to the y-axis can be referred to as a vertical asymptote of the function. Similarly, any asymptote that cross the y-axis parallel to the x-axis is referred to as a horizontal asymptote.

Here is our graph with the vertical and horizontal asymptotes plotted as dashed lines:

Graph 2

The vertical and horizontal asymptotes help us to find the domain and range of the function. We see that the vertical asymptote has a value of x = 1. From this, we can state that the domain of this function consists of all values in x, except for 1. If we look back at our original function, note that the denominator is the term x - 1. Using 1 as a value of x would result in a value of 0 for the denominator. Because division by 0 is undefined, it's mathematically reasonable that the domain of our rational function consists of all values except for 1. In the case of more complex polynomial equations, we can factor the denominator and then solve for 0 in each of the individual terms. Any of those values will be outside the domain of the function, as the function is undefined at those locations.

In a similar manner, we can graphically see that a horizontal asymptote occurs at the value of y = 10. From this, we can state that the range of this function consists of all values in y except for 10.

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