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A reciprocal function can be considered as the quotient where the denominator is a polynomial with a degree of In other words, there is a variable in the denominator. In this lesson, rational functions in which the numerator and the denominator have a degree of at least will be graphed.

Challenge

Average Cost Function

LaShay's school, Jeffereson High, visits a local pencil factory. The average cost, in thousand dollars, of producing thousand pencils is given by the following function.
a Graph the average cost function.
b Identify the domain and range of the function. Use a graphing calculator if necessary.

Discussion

Rational Functions

A rational function is a function that contains a rational expression. Any function that can be written as the quotient of two polynomial functions and is a rational function.

For any values of where the rational function is undefined. These values are excluded from the domain of the function. One example of a rational function is the reciprocal function.
This function has two asymptotes, the axis and the axis. The applet shows the graph of the reciprocal function and some other rational functions.
Graph of several rational functions

Discussion

Discontinuity of Rational Functions

The graph of a rational function can be a smooth continuous curve, or it can have jumps, breaks, or holes. By looking at the graph of a function, functions can be categorized into two groups: continuous or discontinuous.

Concept

Discontinuous - Function

A function is said to be continuous if its graph can be drawn without lifting the pencil. Otherwise, the function is said to be discontinuous.
Continuous and discontinuous functions
A discontinuous function can have holes, jumps, or asymptotes throughout its graph.
Examples of discontinuous functions
A point where a function is discontinuous is called a discontinuity. Function I has a point of discontinuity at Function II has a point of discontinuity at and Function III has a point of discontinuity at

Concept

Point of Discontinuity

A point of discontinuity of a function is a point with the coordinate that makes the function value undefined. A point of discontinuity can also be considered as an excluded value of the function rule.

Removable Discontinuity

If a function can be redefined so that its point of discontinuity is removed, it is called a removable discontinuity. A removable discontinuity may occur when there is a rational expression with common factors in the numerator and denominator. Consider the following rational function.
For this function, is a point of discontinuity because its denominator is when Next, notice that the function can be simplified using a difference of squares.
Simplify right-hand side
After simplifying, the graph behaves as a linear function except in the point of discontinuity, where its value is undefined.
Rational function with removable discontinuity
However, the function can be made continuous by redefining it at If is substituted for in the function it is found that Therefore, the following function is continuous at
This means that is a removable discontinuity. Note that a removable discontinuity is typically a hole in the graph.

Non-Removable Discontinuity

When a function cannot be redefined so that the point of discontinuity becomes a valid input, it is called a non-removable discontinuity. Consider, for example, Since is not in the domain of the function, there is a point of discontinuity at

Rational function with non-removable discontinuity
This is a non-removable discontinuity because there is no way to redefine the function so that it becomes continuous at

Example

Modeling Annual Income Per Capita

LaShay lives in Des Moines, where the total annual amount of personal income in millions dollars, and the population in millions, are modeled by the following functions.
In these functions, represents the number of years that have passed since
a Write a function modeling the annual income per capita in Des Moines.
b Identify the points of discontinuity of , if any.

Hint

a Divide the total amount of personal income by the population. Recall how to divide rational expressions.
b For which values of is the function undefined?

Solution

a Take a look at what the given equations represent.
Total Annual Income Population
In order to write a function for the annual income per person the total annual income must be divided by the population This will give the annual income of a person.
The binomials in the denominator can be multiplied.
b Consider the function written in Part A.
Recall that division by zero is not defined. Therefore, the rational function is undefined when either of the parentheses equals zero.
is undefined
This means that and are excluded values.
If a real number is not in the domain of a function, then the function has a point of discontinuity at Since the domain is all real numbers except and there are two points of discontinuity of the function.

Example

Finding Point of Discontinuity

On the way to the pencil factory, LaShay noticed that road maintenance work is being carried out to fill a hole on the road.

A map that shows a lake, a forrest, some buildings, and a road with a hole in it
The equation below models the road to the factory.
a Find the coordinate of the point where the road maintenance is being completed.
b The municipality of Des Moines plans to build a new road, which is modeled by the following function.
Graph the function.

Answer

a
b Graph:

Hint

a Start by factoring the numerator and the denominator of the function. Find the point of discontinuity of the graph.
b Check if the numerator and the denominator have a common factor.

Solution

a The terms in the numerator of the function have a common factor of Also, the denominator can be factored by grouping.
Notice that there is no value that can make equal zero. However, the other factor in the denominator makes the function undefined when Therefore, this value is excluded from the domain.
Recall that the point of discontinuity occurs at when is not in the domain of a function. Consequently, there is a point of discontinuity at
In the context of the problem, this is the coordinate of the point where the road is being repaired.

Notice that it is also a removable discontinuity because is the common factor in the numerator and denominator.

b Start by factoring the numerator and denominator to see if the function can be simplified.
Factor
The function is undefined when This means that the domain of the function is all real numbers except After simplification, the function will be a linear function except at
Note that Therefore, the graph of the given function is the graph of the function with a hole at
The graph of N(x)

This graph also has a removable discontinuity at because is the common factor in the numerator and denominator of the rational function.

Discussion

Asymptotes of Rational Functions

An asymptote of a graph is an imaginary line that the graph gets close to as goes to plus or minus infinity or a particular number. For example, the graph of the rational function has two asymptotes — the axis and the axis.

Graph of f(x)= 1/x and its asymptotes

Analyzing the diagram, the following can be observed.

  • As approaches infinity and as approaches negative infinity, the value of the function approaches Therefore, or the axis, is a horizontal asymptote of the graph of
  • As approaches the value of the function approaches either positive or negative infinity. Therefore, or the axis, is a vertical asymptote of the graph of
In the coordinate plane below, the asymptotes for three different graphs are shown.
The graphs with asymptotes of three different functions
The applet demonstrates that asymptotes can be not only vertical and horizontal, but also oblique.
Consider a rational function where is a polynomial with a leading coefficient of and a degree of and where is a polynomial with a leading coefficient of and a degree of
To determine the asymptotes of the function, the denominator and the degrees of the polynomials in the numerator and denominator must be analyzed.

Identifying Vertical Asymptotes

To identify the vertical asymptotes, the function should be in its simplest form. That is, if and have common factors, the function should be simplified first. The vertical asymptotes will occur at the zeros of the denominator. Check out the following examples.

Identifying vertical asymptotes

Identifying Horizontal Asymptotes

The degrees of polynomials in the numerator and denominator of a rational function will determine if the graph has a horizontal asymptote.

Horizontal Asymptote If there is no horizontal asymptote.
If the horizontal asymptote is
If the horizontal asymptote is
The graph of a rational function may have one or more vertical asymptotes and at most one horizontal asymptote.
Functions y=x/(x^2-9), y=3x^2/(x^2-x), and y=-x^4/(5x^2-5) are graphed with the vertical and horizontal asymptotes
Notice that the graph of a rational function will never cross a vertical asymptote, whereas the graph may or may not cross a horizontal asymptote.

Example

Identifying Asymptotes

LaShay takes a travel-sickness pill about minutes before the school bus leaves for the field trip. The amount in milligrams of the medication in her bloodstream is modeled by the following rational function.
Here, represents the time in hours after one pill is taken. Identify the asymptotes of the function.

Answer

Vertical Asymptotes: and
Horizontal Asymptote:

Hint

Factor the numerator. What does the Factor Theorem state?

Solution

To identify the asymptotes, the numerator and denominator of the function need to be factored. Notice that the terms in the numerator have a common factor of
To factor the polynomial in the denominator, use the Factor Theorem.

Factor Theorem

The expression is a factor of a polynomial if and only if the value is a zero of the related polynomial function.

Now, the aim is to find all real zeros of the polynomial. Before doing so, identify the degree of the polynomial.
It is a polynomial with a degree of Subsequently, by the Fundamental Theorem of Algebra, it is known know that has exactly three roots. Additionally, by the Rational Root Theorem, integer roots must be factors of the constant term. Since the constant term of is the possible integer roots are and
Three roots for were found, so there is no need for further investigation. In conclusion, and are all the real roots of Next, apply the Factor Theorem to write its factored form.
Now that the factored form of the denominator has been found, the given function can be simplified.
Since the numerator and the denominator share the factor there is a hole at Also, the denominator becomes zero when and These values make the function undefined. Therefore, the function has two vertical asymptotes.
To find the horizontal asymptote, compare the degrees of the polynomials in the numerator and denominator.
Asymptote
none
Look at the degrees of the numerator and denominator of the function.
The degree of the denominator is greater than the degree of the numerator, Therefore, the horizontal asymptote is

Discussion

Oblique Asymptote

An asymptote can be neither vertical nor horizontal. It can be a slanted line. In such cases, it is called an oblique asymptote or slant asymptote.

Graph of f(x)=(x^2+1)/x with a vertical and oblique asymptotes
An oblique asymptote of a rational function occurs when the degree of the numerator is more than that of the denominator. Therefore, a function with an oblique asymptote can never have a horizontal asymptote. To find the equation of the oblique asymptote of a rational function is divided by using long division to get a quotient with a remainder
The equation of the oblique asymptote is of the form where is a non-zero real number.

Example

To illustrate how the technique is used, the asymptotes of the following function will be found.
The difference between the degrees of the numerator and denominator is This means that its graph does not have a horizontal asymptote but an oblique asymptote. To find the oblique asymptote, polynomial long division will be used.
Divide

Multiply by

Subtract down

The quotient is with a remainder of
Therefore, is the oblique asymptote of the rational function.
Graph of f(x) = (x^3+2x)/(4x^2-1)

Example

Modeling Interest in The Trip

The students spent about hours at the factory. At the end of this period, LaShay's teacher made a survey about the trip and drew a graph relating the students' interest in the trip and the time elapsed at the factory. The greater the value, the greater the interest of the students.

The graph of the function S(t) in the first quadrant
The graph above is the part of the graph of the following function in the first quadrant.
Identify the asymptotes of the function.

Answer

Vertical Asymptotes:
Oblique Asymptote:

Hint

Start by subtracting the rational expressions on the right-hand side of the function.

Solution

First, the rational expressions on the right-hand side will be subtracted. The least common denominator of the expressions is Therefore, the first expression needs to be multiplied by so that both have the same denominator.
Simplify right-hand side
Rewrite the denominator
The numerator and the denominator do not have any common factors. Therefore, the function does not have a point of discontinuity, or hole, in its graph. However, it has a vertical asymptote at because the function is undefined at that value.
Now, to determine if the given rational function has a horizontal or oblique asymptote, the degrees of the polynomials in the numerator and the denominator will be compared.
The degree of the numerator is greater than the degree of the denominator, so the graph has no horizontal asymptotes. However, since the degree of the numerator is more than that of the denominator, the graph has an oblique asymptote. Its equation is found by dividing the numerator by the denominator. Use long division to do this.
Divide

Multiply by

Subtract down

Divide