1 Even and odd functions

1.1 Constructing even and odd functions

A given function f ( x ) can always be split into two parts, one of which is even and one of which is odd. To do this write f ( x ) as 1 2 [ f ( x ) + f ( x ) ] and then simply add and subtract 1 2 f ( − x ) to this to give

f ( x ) = 1 2 [ f ( x ) + f ( − x ) ] + 1 2 [ f ( x ) − f ( − x ) ]

The term 1 2 [ f ( x ) + f ( − x ) ] is even because when x is replaced by − x we have 1 2 [ f ( − x ) + f ( x ) ] which is the same as the original. However, the term 1 2 [ f ( x ) − f ( − x ) ] is odd since, on replacing x by − x we have 1 2 [ f ( − x ) − f ( x ) ] = − 1 2 [ f ( x ) − f ( − x ) ] which is the negative of the original.

Example 2

Separate x 3 + 2 x into odd and even parts.

Solution

f ( x ) = x 3 + 2 x

f ( − x ) = ( − x ) 3 + 2 − x = − x 3 + 2 − x

Even part:

1 2 ( f ( x ) + f ( − x ) ) = 1 2 ( x 3 + 2 x − x 3 + 2 − x ) = 1 2 ( 2 x + 2 − x )

Odd part:

1 2 ( f ( x ) − f ( − x ) ) = 1 2 ( x 3 + 2 x + x 3 − 2 − x ) = 1 2 ( 2 x 3 + 2 x − 2 − x )

Task!

Separate the function x 2 − 3 x into odd and even parts.

First, define f ( x ) and find f ( − x ) :

f ( x ) = x 2 − 3 x , f ( − x ) = x 2 − 3 − x Now construct 1 2 [ f ( x ) + f ( − x ) ] , 1 2 [ f ( x ) − f ( − x ) ] :

1 2 [ f ( x ) + f ( − x ) ] = 1 2 ( x 2 − 3 x + x 2 − 3 − x )

= x 2 − 1 2 ( 3 x + 3 − x ) . This is the even part of f ( x ) .

1 2 [ f ( x ) − f ( − x ) ] = 1 2 ( x 2 − 3 x − x 2 + 3 − x )

= 1 2 ( 3 − x − 3 x ) . This is the odd part of f ( x ) .

1.2 The odd and even parts of the exponential function

Using the approach outlined above we see that the even part of e x is

1 2 ( e x + e − x )

and the odd part of e x is

1 2 ( e x − e − x )

We give these new functions special names: cosh x (pronounced ‘cosh’ x ) and sinh x (pronounced ‘shine’ x ).

Key Point 3

Hyperbolic Functions

cosh x ≡ 1 2 ( e x + e − x )
sinh x ≡ 1 2 ( e x − e − x )

These two functions, when added and subtracted, give

cosh x + sinh x ≡ e x and cosh x − sinh x ≡ e − x

The graphs of cosh x and sinh x are shown in Figure 4.

Figure 4 :

The image displays the graphs of the hyperbolic sine function, \text{sinh}(x), and the hyperbolic cosine function, \text{cosh}(x), plotted on a Cartesian coordinate system.

Long description

Note that cosh x > 0 for all values of x and that sinh x is zero only when x = 0 .