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 ?

X | 1 | 10 | 100 |

F(x) | 1 | 0.1 | 0.01 |

X | -1 | -10 | -100 |

F(x) | -1 | -0.1 | -0.01 |

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

- one vertical asymptote
- many vertical asymptotes
- or no vertical asymptotes

The end behavior of rational function around vertical asymptote are:

Figure 7 – As x approaches a, f(x) increases boundless

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:

- The polynomials and have no common factors, if they have common factor must be eliminated.
- 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: