Rational Functions

Rational functions are special functions that you cannot call polynomials, but are obtained by dividing polynomials. In other words, they are the quotients of the polynomial division.

A rational function is of form where and are polynomials and

. The domain of rational function can be any real numbers except those that makes .

Example #1

is a rational function. The function accept all real number except 1. The value 1 makes the function invalid. Hence, we can write domain of the as .

Example #2

Find the domain of rational function .

Solution:

The rational function accepts all values, except , which we can write in interval notation also, .

The Reciprocal Function

The simplest of rational function is the reciprocal function . The function accepts all real values except .

Let us plot the graph of this rational function for following values.

X	-1	–0.5	-0.1	-0.01	-0.001	-0.0001
F(x)	-1	-2	-10	-100	-1000	-10000

As the value approaches from left, the value of becomes smaller and smaller boundlessly to . This can be shown with the arrow notation as follows.

,

What happens when comes closer to from right side.

X	0.0001	0.001	0.01	0.1	0.5	1
F(x)	10000	1000	100	10	2	1

,.

As the value approaches 0 from right , the value of increases boundlessly to positive . This is shown above with arrow notation.

What happens when the value of moves away from , that is, increases or decreases boundlessly ?

When the value increases or decreases boundlessly, then the approaches , but not touching .

This is shown in arrow notation below.

, and , .

Vertical Asymptotes

There are several rational functions, out of which is an interesting one. The graph of this function is reflected across the y-axis.

We can describe the end behavior of this graph in the following manner.

,	,
,	,

The line is called the vertical asymptote of the graph. A rational function can have

1. one vertical asymptote
2. many vertical asymptotes
3. or no vertical asymptotes

The end behavior of rational function around vertical asymptote are:

As the value of approaches the , the increases or decreases without bound.

This increase or decrease in end behavior is useful in study of calculus. We can describe how the value of and the function changes using limits.

The moves closer to from left or right, the end behavior changes is shown in limits below.

How To Locate The Vertical Asymptotes ?

If the rational function have vertical asymptotes, then it can be found easily. We know that where and are two polynomials. There are two conditions to find the vertical asymptotes:

1. The polynomials and have no common factors, if they have common factor must be eliminated.
2. The value must be zero of the polynomial , that is, denominator. If is zero then is the vertical asymptotes.

You can understand this with the help of an example.

Example #3

Find the vertical asymptotes of the rational function:

Solution:

The given equation does not have any common factors, therefore, meet the first condition. The denominator accepts all real numbers except which makes it .

Therefore, is the vertical asymptote in the graph of rational function.

Example #4

Find the vertical asymptote of the rational function: .

Solution:

In the given equation, has a common factor. We reduce the common factor and the equation becomes .

The function accepts all real values, except . Therefore, is the vertical asymptotes.

Example #5

Find the vertical asymptotes for the rational function: .

Solution:

The function has no common factor, but there is no value for which the denominator is . Therefore, the function does not have a vertical asymptote.

In some cases, the denominator is shows that it has a zero, but after reducing the common factors, the resultant expression has a totally new vertical asymptote.

Example #6

Find the vertical asymptote for the equation: .

Solution:

At first we see that the equation has a zero , but when the equation is reduced after reducing the common factors, we get and the vertical asymptote is .

Horizontal Asymptotes

The equation represents “vertical asymptote“, similarly, represents the “horizontal asymptotes“. There may be several vertical asymptotes, but there is only one horizontal asymptote.

The function , as increases or decreases without bound or , the approaches , which is . So, we can say that the horizontal asymptote is that which is defined as .

We can write them in limit form as: