Even and Odd Parts

In[9]:=

  Clear[f,f1,f2]

Define two new functions in terms of this function.

In[10]:=

  f1[x_]:=1/2 (f[x]+f[-x])
  f2[x_]:=1/2 (f[x]-f[-x])

Let's look at a few things.

In[11]:=

  Simplify[f1[x]+f2[x]]

Out[11]=

  f[x]

In[12]:=

  f1[-x]
  f2[-x]

Out[12]=

  f[-x] + f[x]
  ------------
       2

Out[13]=

  f[-x] - f[x]
  ------------
       2

Notice anything?

In[14]:=

  Simplify[f1[x]-f1[-x]]

Out[14]=

  0

In[15]:=

  Simplify[f2[x]+f2[-x]]

Out[15]=

  0

Hmm...

The function 1/2 [f(x)+f(-x)] is called the even part of f(x).
The function 1/2 [f(x)-f(-x)] is called the odd part of f(x).

Examples