Rational expression

Rational expression:

An expression which contains polynomials in the numerator and/or in the denominator is called rational expressions.

Example:

`(3x+1)/(x-1)` is a rational expression.

Rational function:

A rational function is a function y = f(x) where f( x) represents a rational expression.

Therational functions is graphed by finding the values of the function forwhich the function is undefined. A function will become undefined for the values that make the denominator zero. Dashed lines are drawn on thegraph for any values for which the rational function is undefined. These lines are called asymptote lines. The graph of the rational function will get close to these asymptote lines but will never intersect them.


Steps to find the rational function


1.First, we have to find the domain for the given function. To find the domain of the function. we have to set each factor to the denominator equal to zero. Find the function where the function is undefined. The domain of the function will be all real numbers except the numbers in which the function becomes undefined.

2.Secondly, we have to find the vertical asymptote. To find the vertical asymptote we have to simplify the factored expression. Set the remainingfactors of the denominator to zero and find the values of x. This valueof x will give the vertical asymptote.

3. In the next step we have to find the horizontal asymptote. To find the horizontal asymtote we have to follow the below mentioned procedure. It depends on the largest exponents of the variable in the numerator and the denominator.
(i)If the degree (highest power) of the numerator is larger than the degree of the denominator, then there is no horizontal asymptote.
(ii)If the degree of the numerator is smaller than the degree of the denominator, then the horizontal asymptote is at  y = 0  (the x–axis).

(iii)If the degree of the numerator is equal to the degree of the denominator, then compare the coefficients in front of the terms with the highest power. The horizontal asymptote is the coefficient of the highest power of the numerator divided by the coefficient of the highestpower of the denominator.

4. Finally, plot the vertical line at the vertical asymptote and draw a horizontal line at the horizontal lineand graph the rational function.


Example


Graph the expression `(5x+1)/(5x-1)`

Solution:

First, we have to find the domain of the function.<br> 

Equate the denominator to zero and find the value of x.

5x - 1 = 0 ===>  x = 1/5

So, the domain of the function is all real numbers except  1/5.

(ii) Secondly, we have to find Vertical asymptote.

Since there is no common term between the numerator and denominator we get the vertical asymptote as x = 1/5.

(iii) Horizontal asymptote:

Since the exponent of the numerator and denominator are same. Hence, 

y =1 is the equation.

Graph: