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

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^{2 }+ 2x - 3) / (x^{2 }- 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) = (x^{2} - 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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