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).

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

You can also visit the following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**