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. 

The sum or difference of two even functions is always even. 

The sum or difference of two odd functions is always odd. 

Application

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) = x3, 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) = x3

f(-x) = (-x)3

= -x3

= -f(x)

Since f(-x) = -f(x),

f(x) is an odd function

Example 2 :

Let f(x) = x2  + 2, is f(x) odd or even function?

Solution :

f(x) = x2  + 2

f(-x) = (-x)2 + 2

= x2 + 2

= f(x)

Since f(-x) = f(x),

f(x) is an even function

Example 3 :

Let f(x) = x3 - 2x, is f(x) odd or even function?

Solution :

f(x) = x3 - 2x

f(-x) = (-x)3 - 2(-x)

= -x3 +  2x

= -(x3 - 2x)

= -f(x)

Since f(-x) = -f(x),

f(x) is an odd function

Example 4 :

Let f(x) = 5x3 + x2 - 1, is f(x) odd or even function?

Solution :

f(x) = 5x3 + x2 - 1

f(-x) = 5(-x)3 + (-x)- 1

= 5(-x3) - x- 1

= -5x3 - x- 1

= -(5x3 + x+ 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) = x4 + 2x2  + 5, is f(x) odd or even function?

Solution :

f(x) = x4 + 2x2  + 5

f(-x) = (-x)+ 2(-x)2 + 2

= x+ 2x2 + 2

 = f(x)

Since f(-x) = f(x),

f(x) is an even function

Important Note :

In trigonometric ratios, if we have negative angle, we have to understand that the angle will fall in the IVth quadrant.

In IVth quadrant, the trigonometric ratios 'cos' and 'sec' are positive and all other trigonometric ratios are negative.

To know more about trigonometric ratios in all the four quadrants, click here.

Example 6 :

Is sinx odd or even function?

Solution :

Let f(x) = sinx.

f(x) = sinx

f(-x) = sin(-x)

Because the angle is negative, it falls in the IVth quadrant. In IVth quadrant sine is negative.

f(-x) = -sinx

= -f(x)

Since f(-x) = -f(x),

f(x) is an odd function

Therefvore, sinx is an odd function.

Example 7 : 

Is cosx odd or even function?

Solution :

Let f(x) = cosx.

f(x) = cosx

f(-x) = cos(-x)

In IVth quadrant cosine is positive.

f(-x) = cosx

= f(x)

Since f(-x) = f(x),

f(x) is an even function

Therefore, cosx is even function.

Example 8 : 

Is tanx odd or even function?

Solution :

Let f(x) = tanx.

f(x) = tanx

f(-x) = tan(-x)

In IVth quadrant tangent is negative.

f(-x) = -tanx

f(-x) = -f(x)

Since f(-x) = -f(x),

f(x) is an odd function

Therefore, tanx is odd function.

Example 9 : 

Is secx odd or even function?

Solution :

Let f(x) = secx.

f(x) = secx

f(-x) = sec(-x)

In IVth quadrant secant is positive.

f(-x) = secx

f(-x) = f(x)

Since f(-x) = f(x),

f(x) is an even function

Therefore, secx is even function.

Example 10 :

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

Solution :

f(x) = sinx + tanx

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

f(-x) = -sinx - tanx

= -(sinx + tanx)

= -f(x)

Since f(-x) = -f(x),

f(x) is an odd function

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

Example 11 : 

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

Solution :

f(x) = secx + cosx

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

f(-x) = secx + cosx

f(-x) = f(x)

Since f(-x) = f(x),

f(x) is an even function

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

Example 12 : 

If the point (8, -3) is on the graph of y = f(x), name another point on the graph if 

(a) f(x) is an even function

(b) f(x) is an odd function 

Solution :

Since the point (8, -3) is on the graph of y = f(x),

x = 8, y = -3

When x = 8, the value of f(x) is -3.

f(8) = -3 ----(1)

Part (a) :

Given : f(x) is an even function.

f(-x) = f(x)

f(-8) = f(8)

f(-8) = -3 ........from (1)

Another point on the graph is (-8, -3).

Part (b) :

Given : f(x) is an odd function.

f(-x) = -f(x)

f(-8) = -f(8)

f(-8) = -(-3) ........from (1)

f(-8) = 3

Another point on the graph is (-8, 3).

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