Page 130 - DMTH401_REAL ANALYSIS

Real Analysis

Notes 9.4.6 Even and Odd Functions
Consider a function f : R  R defined as

f(x) = 2x, " x R.
If you change x to –x, then you have
f(–x) = 2 (–x) = –2x – f(x).

Such a function is called an odd function.
Now, consider a function f : R  R defined as
f(x) = x " x R
2
Then changing x to –x we get
2
2
q – x) = (–x) = x = f(x)
Such a function is called an even function.
The definitions of even and odd functions are as follows:
A function f : R  R is called even if f (–x) = f(x), " x R,

It is called odd if f(–x) = –f(x), " x R

Example: Verify whether the function f : R  R defined by

3
2
(i) f(x) = Sin x + Cos 2x
2
2
2
2

(ii) f(x) = a  ax x  a  ax x are even or odd.

Solution:
2
3
(i) f(x) + sin x + cos 2x, " x R
3
 sin (–x) + cos 2(–x)
2
3
sin x + cos 2x = f(x), " x R
2
 f is an even function.
2
2
2
2


(ii) f(x) = a  ax x  a  ax x , x R
"

2
2
2


 f(–x) = a  ax x  a  ax x 2
= –(x), " x R
 f is an odd function.
Task Determine which of the following functions are even or odd or neither:
(i) f(x) = x
(ii) A constant function
(iii) sin x, cos x, tan x,

x 4
(iv) f(x) = = –(x), " x R
2
x  9
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