HOW TO FIND THE HOLE OF A RATIONAL FUNCTION

In this section, you will learn 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

So, 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).

Example 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 :

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

Example 2 :

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

f(x)  =  (x+ 2x - 3)/(x- 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.

Example 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

So, 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 substitute 2 for x, we get get

f(2)  =  3

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

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