function

Even and Odd Functions
Even Function:
A functions is said to be an even function if .
Graph of an even function is symmetrical about the -axis, i.e., if point lies on the graph then also li
graph.

Graph:

, Odd function:
A function is said to be an odd function if .
Graph of an odd function is symmetrical in opposite quadrants, i.e., if point lies on the graph then also li
graph.

Graph:




Properties of Odd and Even functions:
• Sometimes, it is easy to prove that for even functions and for odd functions.
• A function can be either even or odd or neither.
• Function (not necessarily even or odd) can be expressed as a sum of an even an odd function. i.e.,

Let and . It can now easily be shown that is even and is odd.

,• The first derivative of an even function is an odd function and vice versa.
• If domain of , then for odd function which is continuous at , i.e., if for a function, , then that function cannot b
follows that for a differentiable even function . i.e., if for a differentiable functions then the function cannot be
• is the only function which is defined on the entire number line is even and odd at the same time.
• Every even function is many-one .




Even Even Even Even Even Even Eve

Even Odd Neither even nor odd Neither even nor odd Odd Odd Eve

Odd Even Neither even nor odd Neither even nor odd Odd Odd Eve

Odd Odd Odd Odd Even Even Odd

, Illustration graph the function:

Which of the following functions is (are) even, odd or neither

(a) . (b) .