Properties of Rational Function

A rational function has various properties which are as follows:

• Domain of Rational Function
• Range of Rational Function
• Asymptotes of Rational Function
  • Horizontal Asymptote
  • Vertical Asymptote
  • Oblique Asymptote
• Holes of Rational Function

Domain of Rational Function

We know that the denominator is never zero in a rational function. This fact is used to find out the domain and range of the rational function. In order to calculate the domain of the rational function following steps are followed

• Keep the value of denominator equal to zero
• Find the values of variable where the denominator becomes zero.
• The domain of the function is thus the set of all real numbers R excluding the values which make the denominator zero.

Range of Rational Function

In order to calculate the range following steps are followed:

• Set f(x)=y
• Solve the equation so obtained for the variable x.
• Now use the condition of denominator nit equals to zero.
• Find the value of y from the above condition.
• The range of the function is set of all real number R excluding the value of y so obtained.

Let’s consider an example for understanding the domain and range of Rational Function.

Example: Consider a rational function (x+2)/(x+1), and find its domain and range.

Solution:

To calculate the domain of the given function, we can follow the following steps:

• Equate the denominator to zero i.e. x+1=0, which gives x=-1.
• Thus the domain of the function is R-{-1}.

The range of the function is calculated as follows:

• Set f(x)=y=(x+2)/(x+1)
• Solve the equation for x

y(x+1) = x+2

⇒ xy+y = x+2

⇒ xy-x = 2-y

⇒ x(y-1) = 2-y

⇒ x = 2-y/(y-1)

• Keep denominator not equals to zero i.e. y-1≠0 which gives y≠1.
• Thus range of the function is set of real numbers R-{1}.

Asymptotes of Rational Function

All the Rational Functions have three types of Asymptotes and they are,

• Vertical Asymptote
• Horizontal Asymptote
• Oblique Asymptote

Now, let’s learn about them in detail

Vertical Asymptote

A vertical asymptote of a rational function is a line parallel to the y-axis and is of the form x=a where a is any number. This line appears to touch the graph of the rational function but it never touches it actually. There can be one or more vertical asymptote of a rational function. In order to find the vertical asymptote of the rational function, following steps are followed:

• Reduce the rational function to its lowest form and eliminate all the common terms in numerator and denominator.
• Set the denominator equal to zero and find the value of the variable for which denominator becomes zero.

Horizontal Asymptote

A horizontal asymptote of a rational function is a line parallel to the x-axis and is of the form y=a where a is any number. This line appears to touch the graph of the rational function but it never touches it actually. There can be only one horizontal asymptote of a rational function. Horizontal asymptote of a rational function can be calculated by comparing the degrees of numerator and denominator as follows:

• Let N and D be the degree of numerator and denominator respectively.
• If N < D, horizontal asymptote is y=0
• If N > D, then there is no horizontal asymptote
• If N = D, y = ratio of leading coefficients in numerator and denominator is the horizontal asymptote.

Oblique Asymptote

It is a slant line which appears to touch the graph of the rational function but never touches it. It is present only in the case where degree of numerator N = degree of denominator D+1. The horizontal asymptote is equal to the quotient which is the result of division of numerator and denominator of the rational function.

Holes of Rational Function

The points that appear to be present on the graph of the rational function but are actually not present are called holes of rational function. In order to calculate the holes, first reduce the function to lowest form and set the common factor equal to zero. The value of the variable and value of function at that variable is the required hole.

Rational Function is a type of function that is expressed as a fraction where both the numerator and denominator must be a polynomial and the denominator can never equal zero. Thus a rational function is similar to a fraction but the numerator and denominator are polynomial functions. In simple words, the rational function can be defined as the ratio of two polynomials. Rational functions find applications in various daily life problems and in various fields in life.

In this article, we shall discuss rational function in detail.