**Even and Odd Functions :**

Let f(x) be a function.

To find whether f(x) is even or odd, we have to replace "x" by "-x" in f(x). We have to conclude f(x) as even or odd function from the result of f(-x) as shown below.

**1. If f(-x) = f(x), then f(x) is even function**

**2. If f(-x) = - f(x), then f(x) is odd function**

If f(-x) is neither equal to f(x) nor -f(x), we have to conclude f(x) is neither even nor odd.

Even though we have many applications of even and odd functions, let us consider the important application of even and odd functions in integral calculus.

If f(x) is even function,

If f(x) is odd function,

**Example 1 : **

Let f(x) = x^{3}, is f(x) odd or even function ?

**Solution : **

To know f(x) is odd or even function, let us plug x = -x in f(x).

Then, we have

f(-x) = (-x)^{3}

f(-x) = -x^{3}

f(-x) = - f(x)

So, f(x) is odd function.

**Example 2 : **

Let f(x) = x^{2} + 2, is f(x) odd or even function ?

**Solution : **

To know f(x) is odd or even function, substitute -x for x in f(x).

Then, we have

f(-x) = (-x)^{2} + 2

f(-x) = x^{2} + 2

f(-x) = f(x)

So, f(x) is even function.

**Example 3 : **

Let f(x) = x^{3} - 2x, is f(x) odd or even function ?

**Solution : **

To know f(x) is odd or even function, substitute -x for x in f(x).

Then, we have

f(-x) = (-x)^{3} - 2(-x)

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

f(-x) = -(x^{3} - 2x)

f(-x) = -f(x)

So, f(x) is odd function.

**Example 4 : **

Let f(x) = 5x^{3} + x^{2 }- 1, is f(x) odd or even function ?

**Solution : **

To know f(x) is odd or even function, substitute -x for x in f(x).

Then, we have

f(-x) = 5(-x)^{3} + (-x)^{2 }- 1

f(-x) = 5(-x^{3}) - x^{2 }- 1

f(-x) = -5x^{3} - x^{2 }- 1

f(-x) = -(5x^{3} + x^{2 }+ 1)

f(-x) can not be expressed as either as f(x) or -f(x).

So, f(x) is neither even nor odd function.

**Example 5 : **

Let f(x) = x^{4} + 2x^{2} + 5, is f(x) odd or even function ?

**Solution : **

To know f(x) is odd or even function, substitute -x for x in f(x).

Then, we have

f(-x) = (-x)^{4 }+ 2(-x)^{2} + 2

f(-x) = x^{4 }+ 2x^{2} + 2

f(-x) = f(x)

So, f(x) is even function.

**Important Note : **

In trigonometric ratios, if we have negative angle, we have to understand that the angle will fall in the IV^{th }quadrant.

In IV^{th} quadrant, the trigonometric ratios "cos" and "sec" are positive and all other trigonometric ratios are negative

**Example 6 : **

Is sinx odd or even function ?

**Solution : **

Let f(x) = sinx

To know f(x) is odd or even function, substitute -x for x in f(x).

Then, we have

f(-x) = sin(-x)

Because the angle is negative, it falls in the IV^{th} quadrant. In IV^{th} quadrant "sin" is negative.

So, we have

f(-x) = - sinx

f(-x) = - f(x)

f(x) is odd function

So, sinx is odd function.

**Example 7 :**** **

Is cosx odd or even function ?

**Solution : **

To know f(x) is odd or even function, substitute -x for x in f(x).

Then, we have

f(-x) = cos(-x)

Because the angle is negative, it falls in the IV^{th} quadrant. In IV^{th} quadrant "cos" is positive.

So, we have

f(-x) = cosx

f(-x) = f(x)

f(x) is even function

So, cosx is even function.

**Example 8 :**** **

Is tanx odd or even function ?

**Solution : **

Let f(x) = tanx

To know f(x) is odd or even function, substitute -x for x in f(x).

Then, we have

f(-x) = tan(-x)

Because the angle is negative, it falls in the IV^{th} quadrant. In IV^{th} quadrant "tan" is negative.

So, we have

f(-x) = - tanx

f(-x) = - f(x)

f(x) is odd function

So, tanx is odd function.

**Example 9 :**** **

Let f(x) = sinx + tanx, is f(x) odd or even function ?

**Solution : **

To know f(x) is odd or even function, substitute -x for x in f(x).

Then, we have

f(-x) = sin (-x) + tan(-x)

Because the angle is negative, it falls in the IV^{th} quadrant. In IV^{th} quadrant both "sin" and "tan" are negative.

So, we have

f(-x) = - sinx - tanx

f(-x) = - (sinx + tanx)

f(-x) = - f(x)

So, f(x) is odd function.

**Note : **The sum or difference of two odd functions is always odd.

**Example 10 :**** **

Let f(x) = secx + cosx, is f(x) odd or even function ?

**Solution : **

To know f(x) is odd or even function, substitute -x for x in f(x).

Then, we have

f(-x) = sec(-x) + cos(-x)

Because the angle is negative, it falls in the IV^{th} quadrant. In IV^{th} quadrant both "sec" and "cos" are positive.

So, we have

f(-x) = secx + cosx

f(-x) = f(x)

So, f(x) is even function.

**Note : **The sum or difference of two even functions is always even.

After having gone through the stuff given above, we hope that the students would have understood even and odd functions.

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 our 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**