About the topic "Domain and range of rational functions"

"Domain and range of rational functions" 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 functions", we have given step by step explanation.

To learn, "What are domain and range of rational-functions ?", we have to be knowing the following important facts.

Let y = f(x) be any function.

                        Domain (y) = Range (y⁻¹)

                        Range (y) = Domain (y⁻¹)

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 :

Let y = 1 / (x-2)

In the above rational function, if we plug x = 2, the denominator becomes zero. So, "y" is undefined.

Hence, "y" is defined for all real values of "x" except x = 2

Hence, Domain (y)  = R - {2}

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")

Example :

Let y = 1/ (x-2)

To find range of the rational function above, first we have to find inverse of "y".

To find inverse of "y", follow the steps given below.

Step 1:

y = 1/(x-2) has been defined by "y" in terms "x". 

The same function has to be redefined by "x" in terms of "y".

Step 2:

                       y = 1/(x-2) ===> y(x-2) = 1

                                         ===> xy - 2y = 1

                                         ===> xy = 2y + 1

                                         ===> x = (2y + 1)/y

Now the function has been defined by "x" in terms of "y".

Step 3:

In x = (2y+1)/y, we have to replace "x" by y⁻¹ and "y" by "x"

Then we will get,  y⁻¹ = (2x+1)/x

Step 4:

Now, find the domain of y⁻¹.

In the inverse function y⁻¹, if we plug x = 0, the denominator becomes zero. So y⁻¹ is undefined.

Hence, y⁻¹ is defined for all real values of "x" except x = 0.

Hence, Domain ( y⁻¹) = R - {0}

And we already know the fact that Range (y) = Domain (y⁻¹)

Therefore Range (y) = R - {0}

Another way to find range of rational functions

For some rational functions, 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.

In this way, we can easily get the domain and range of rational functions from the graphs.

Let us see, how we can find the range of a rational function from its graph.

Vertical Asymptote:

To find vertical asymptote, we have to make the denominator of         y = 1/ (x-2) equal to zero.

When we do so,    x - 2 = 0  ====>    x  =  2

So, the vertical asymptote is   x  =  2

Horizontal Asymptote:

In the rational function y = 1/ (x-2), the highest exponent of the numerator is less than the highest exponent of the denominator. 

So there is an horizontal asymptote and it is y =0

Table of Values :

In the given rational function y = 1/(x-2), now we have to plug some random values for "x" and find the corresponding values of "y". 

We have already known that the vertical asymptote is x = 2

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

x<2, x>2 but not x=2.

(Because, x = 2 is vertical asymptote)

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                          y = 1/(x-2) appears at every real value of "y" except y = 0.

From the graph, clearly Range (y)  = R - {0}

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

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