HOW TO FIND THE HOLE OF A RATIONAL FUNCTION

About the topic "How to find the hole of a rational function"

"how to find the hole of a rational function ?" is the question having had by the students who are studying math in school final. Even though it is taught by the teachers in school and university, students do not understand this clearly. Often students have this question on slant asymptotes. 

On this page of our website, we have given step by step explanation and examples to make the students to clearly understand how to how to find the hole of a rational function.  

And we will be able to find the hole of a function, only if it is a rational function.

That is, the function has to be in the form of f(x) = P/Q

Example : Rational Function

Steps involved in finding hole of a rational function

Let y = f(x) be the given rational function. 

Step 1 :

If it is possible, factor the polynomials which are found at the numerator and denominator.  

Step 2 :

After having factored the polynomials at the numerator and denominator, we have to see, whether there is any common factor at both numerator and denominator. 

Case 1 :

If there is no common factor at both numerator and denominator, there is no hole for the rational function. 

Case 2 :

If there is a common factor at both numerator and denominator, there is a hole for the rational function.

Step 3 :

Let (x-a) be the common factor found at both numerator and denominator.

Now we have to make (x-a) equal to zero.

When we do so, we get

x-a = 0

x = a

Hence, there is a hole at x = a 

Step 4 :

Let y = b for x = a

So, the hole will appear on the graph at the point ( a , b )

Examples:

1. Find the hole (if any) of the function given below

f(x) = 1 / (x+6)

Solution :

Step 1:

In the given rational function, clearly there is no common factor found at both numerator and denominator. 

Step 2 :

Hence, there is no hole for the given rational function.

2. Find the hole (if any) of the function given below

f(x) = (x²+2x-3) / (x²-5x+6)

Solution :

Step 1:

In the given rational function, let us factor the numerator and denominator.

f(x) = [(x+3)(x-1)] / [(x-2)(x-3)]

Step 2 :

After having factored, there is no common factor found at both numerator and denominator. 

Step 3 :

Hence, there is no hole for the given rational function.

3. Find the hole (if any) of the function given below

f(x) = (x² -x - 2) / (x-2)

Solution :

Step 1:

In the given rational function, let us factor the numerator .

f(x) = [(x-2)(x+1)] / (x-2)

Step 2 :

After having factored, the common factor found at both numerator and denominator is (x-2)

Step 3 :

Now, we have to make this common factor (x-2) equal to zero.

x - 2 = 0

x = 2

Hence, there is a hole at x = 2

Step 4 :

After crossing out the common factors at both numerator and denominator in the given rational function, we get

f(x) = x+1  -------(1)

If we plug x = 2 in (1), we get get f(2) = 3

So, the hole will appear on the graph at the point (2,3)




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