# DOMAIN AND RANGE OF RATIONAL FUNCTION GRAPHS

## About the topic "Domain and range of rational function graphs"

"Domain and range of rational function graphs" is a much needed stuff required by almost all the students who study math in high schools.

Even though students can get this stuff on internet, they do not understand exactly what has been explained.

To make the students to understand the stuff "Domain and range of rational function graphs", we have given step by step explanation.

## What is domain?

Let y = f(x) be a function.

Domain is nothing but the real values of "x" for which "y" is defined.

Example :

In the above rational function, let us make the denominator equal to zero.

That is,                              x² - 5x -6 = 0

(x-6)(x+1) = 0

x -6 = 0 or x + 1 = 0

x = 6 or x = -1

In the rational function given above, the denominator becomes zero when x = 6 and x = -1.

Hence, "y" is defined for all real values of "x" except x = 6 and                x = -1

Hence, Domain (y)  = R - {-1,6}

## What is range?

Let y = f(x) be a function.

Range is nothing but the real values of "y" for the given domain (real values of "x")

To find range of the function f(x) = (x² + 2x -3)/(x² - 5x -6),  first we have to find inverse of "f(x)".

But, for some rational functions, it is bit difficult to find inverse function.

For our rational function f(x) = (x² + 2x -3)/(x² - 5x -6) also, it is bit difficult to find inverse function.   In that case, we have to sketch the graph of the rational function using vertical asymptote, horizontal asymptote and table of values as given below.

Vertical Asymptote:

To find vertical asymptote, we have to make the denominator of         f(x) = (x² + 2x -3)/(x² - 5x -6) equal to zero.

When we do so, we get,

x² - 5x -6 = 0 ===> x = -1 and x = 6.

So, the vertical asymptotes are  x  =  -1 and x = 6

Horizontal Asymptote:

In the rational function f(x) = (x² + 2x -3)/(x² - 5x -6), the highest exponent of the numerator and denominator are equal and it is "2".

Equation of the horizontal asymptote:

y = a/b

Here "a" and "b" are the coefficients of highest exponent terms at the numerator and denominator respectively.

In our problem,   a  =  1  and b  =  1

So, horizontal asymptote is y = 1/1 ==> y  =  1

Table of Values :

In the given rational function, now we have to plug some random values for "x" and find the corresponding values of "y".

We have already known that the vertical asymptotes are x = -1  and x = 6.

Now, we have to take some random values for x in the following intervals.

x<-1, -1<x<6,  x>6 but not x = -1 & x = 6.

(Because, x = -1 and x = 6 are vertical asymptotes)

## Graph of the given rational function

When we look at the above graph, the following point is very clear.

That is, "The graph (in red color) of the rational function                     f(x) = (x² + 2x -3)/(x² - 5x -6) appears at every real value of "y" .

(Even though horizontal asymptote is y =1, the graph appears at y = 1. So y = 1 is also included in range)

So the range is all real values.

From the graph, clearly Range (y)  = R

In this way we can find domain and range of rational function graphs easily.

You can also visit the following sites to know more about domain and range of rational functions.

http://hotmath.com

http://www.analyzemath.com

https://cims.nyu.edu

http://www.montereyinstitute.org