Rational Functions

Introduction

A rational function is a function that has a ratio of polynomials. Any function with one variable is called a rational function. 

It is represented as 

f (x) = p (x) / q (x), 

where p (x) and q (x) are polynomials and q (x) is not zero.

It is easy to find a rational function—if the numerator or denominator is not a polynomial, the fraction does not represent rational functions. One thing you should know while graphing rational functions are asymptotes. The study material and notes on rational functions below can help you understand the concept.

What is the domain of the rational function?

When the denominator is zero, it is not a fraction. It is the key point that helps find the domain of any rational function. The domain of a rational function is the set of all x-values that a function takes. To find the domain:

y = f (x)

First, set the denominator not equal to 0 and then solve it for x.

Set all real numbers other than the value of x mentioned in the last step of the domain. 

Let’s look at an example.

Find the domain of f (x) = (2x + 1) / (3x − 2)

The denominator should not be zero.

3x − 2 ≠ 0

x ≠ 2/3

Thus, the domain is {x ∈ R | x ≠ 2/3}. 

What is the range of the rational function?

The rational number range is a set of all y value outputs produced. To find range:

y = f (x)

First, replace f (x) with y.

Solve the equation. 

Set the denominator to not zero and solve it for y. 

Set all real numbers other than the value of y mentioned in the last step of the range.

Let’s look at an example.

Find the range of f (x) = (2x + 1) / (3x − 2).

Firstly, replace f (x) with y, so now the equation is y = (2x + 1) / (3x − 2)

(3x − 2) y = (2x + 1)

Or, 3xy – 2y = 2x + 1

Or, 3xy – 2x = 2y + 1

Or, x (3y − 2) = (2y + 1)

Or, x = (2y + 1) / (3y − 2)

Now (3y − 2) ≠ 0

Or, y ≠ ⅔

So the range is {y ∈ R | y ≠ 2/3}

What is an asymptote?

A straight line in which a curve approaches it closely and goes to infinity is called an asymptote. In other words, an asymptote of a curve is a line in which the distance between the curve and line approaches zero, and they tend to extend till infinity. A rational function is mostly a horizontal asymptote or slant asymptote. A rational function has three types of asymptotes:

• Horizontal 
• Vertical 
• Slant 

Horizontal

A horizontal asymptote is an imaginary horizontal line on the graph that appears close but never touches. It is in the form of y = some number. This number is connected to excluded values from a range. Mostly, there is only one horizontal asymptote in a rational function. An easy way to find horizontal asymptotes is to use the degree of numerator and denominator.

If the numerator is less than the denominator, then the horizontal asymptote at y = 0.

If the numerator is greater than the denominator, there is no horizontal asymptote.

If the numerator and denominator are equal then, y is equal to the ratio of the reading efficient. 

For example, we have to find the horizontal asymptote for f (x) = (x2 + 5x + 6) / (x2 + x − 2)

Here, the degree of numerator and denominator is the same, 2.

So HA = (leading coefficient of numerator) / (leading coefficient of the denominator)

That is 1/1 = 1 

Therefore, the value of horizontal asymptote is 1.

Vertical 

A vertical asymptote is an imaginary vertical line on the graph that appears close but never touches it. It is a form of x = some number. Some numbers connect to excluded domain values. There can not be a vertical asymptote if a hole is in the same number. A rational function can have various vertical asymptotes. 

To find a vertical asymptote, simplify the functions and cancel all common factors. Set the denominator to zero and solve (x). For instance, we have to find the vertical asymptote for the function

(x) = (x2 + 5x + 6) / (x2 + x − 2).

This function can be simplified to f (x) = (x + 3) / (x − 1).

Setting the denominator to 0, 

x – 1 = 0

x = 1

Thus, the vertical asymptote of this rational function, x = 1 

Slant 

A slant asymptote is also known as an oblique asymptote. It is an imaginary line on the graph that appears to touch it. The rational function has an oblique asymptote only when the degree of the numerator is greater than the degree of the denominator. 

For example, find the slant asymptote for

 F (x) = x2/(x+1).

Here the degree of the numerator is two, and the denominator is 1.

So, let’s divide x2 by (x + 1)

After dividing, the result is y= x – 1

Thus, the slant asymptote for this function is y = x – 1.

Conclusion 

In short, a rational function is a ratio of polynomials whose form is f (x) = P(x) / Q(x), where Q (x) is not 0. Every rational function has at least one vertical asymptote and multiple horizontal asymptotes. The linear factors get cancelled when the rational function is simplified and holes. We can use rational functions in our daily life. These rational equations and formulas help us calculate speed, distances, the work rate of people or machines, and we can also solve mixing problems.