Even & Odd Functions 


A function  g  is said to be even, provided that   g(–x) = g(x),    for all x.   

Such functions have graphs that are symmetric about the y-axis: 

Also any function with a graph symmetric about the y-axis is an even function.   Next we'll look at some examples of even functions and their graphs.


Example 1         

This is an even function since  .   This function’s graph: 


Example 2       

This function is even since 

.

 This function’s graph:


From the above two examples you may have realized that any polynomial with only even exponents is an even function.   The cosine is also an even function...


Example 3       c(x) = cos x 

This function is even since  c(-x) = cos(-x) = cos(x) = c(x).     (This is a trigonometric identity)

The graph of the cosine function:
 


A function f is said to be odd, provided that    f(–x) = –f(x),    for all x.  

Such functions have graphs that are symmetric about the origin:

 

Also any function with a graph symmetric about the origin is an odd function.   Next we'll look at some examples of odd functions and their graphs. 


Example 4        

This function is an odd  function since  .     This function’s graph: 


Example 5          

This function is odd since

 

This function’s graph:


From the above two examples you may have realized that any polynomial with only odd exponents is an odd function.   The sine is also an odd function...


Example 6       s(x) = sin x 

This function is odd since  s(-x) = sin(-x) = -sin(x) = s(x).     (This is a trigonometric identity) 

The graph of the sine function: 


Every function can be written uniquely as the sum of an even function and an odd function.   The even part of f is denoted by  fe  and the odd part by  fo . 

The even and odd parts of f are defined by the equations: 

      and     . 

Then fe  is an even function since

and  fo  is an odd function since

 Furthermore, 


Since not every function is purely even or odd, and yet every function may be written as the sum of an even function and an odd function, it follows that the sum of an even function and an odd function is neither even nor odd.


Example 7     

However, the product of an even function with an odd function is an odd function.  To see why, let g be an even function and h and odd function.  Recall that this means that g(–x) = g(x) and h(–x) = –h(x).   Let  

P(x) = g(x) h(x)

Then   

P(–x) = g(–x) h(–x) 


by definition
                    = g(x) (–h(x)) since g is even and h is odd
                    = – g(x) h(x)  a pos times a neg is a neg
                    = –P(x)  by definition of P

And so P is indeed an odd function.      Notice how this is quite different than the situation with even and odd integers, as the product of an even integer with an odd one results in an even integer.


Example 8

Suppose g is even and h odd, once again.   Let  C be the composition of g with h,

C(x) = g( h(x) )

To see if C is even, odd, or neither, we again compute:

C(–x) = g( h(–x) )


by definition
                    = g( –h(x) ) since h is odd
                    =  g( h(x) )  since g is even
                    = C(x)  by definition of C

Hence C is an even function.


By similar calculations you can decide whether various algebraic combinations of even functions with even and/or odd functions results in a function that is even, odd, or neither.


Example 9

Suppose g is even and h odd.   Let  f  be defined by

Is f even, odd, or neither?

As always, with this type of problem, we can only discover whether  f is even or odd or neither, by plugging –x into the function and seeing what happens:

We have shown that    f(–x) =  f(x),   i.e.,  f is an even function.


   

© 1998-2003 by
Rafael Espericueta
all rights are reserved