It is easy to find the nature of function resulting from mathematical operations, provided we know the nature of operand functions. As already discussed, we check for following possibilities :
 If f(x) = f(x), then f(x) is even.
 If f(x) = f(x), then f(x) is odd.
 If above conditions are not met, then f(x) is neither even nor odd.
Based on above algorithm, we can determine the nature of resulting function. For example, let us determine the nature of "fog" function when “f” is an even and “g” is an odd function. By definition,
f
o
g

x
=
f
g

x
f
o
g

x
=
f
g

x
But, “g” is an odd function. Hence,
⇒
g

x
=

g
x
⇒
g

x
=

g
x
Combining two equations,
⇒
f
o
g

x
=
f

g
x
⇒
f
o
g

x
=
f

g
x
It is given that “f” is even function. Therefore, f(x) = f(x). Hence,
⇒
f
o
g

x
=
=
f

g
x
=
f
g
x
=
f
o
g
x
⇒
f
o
g

x
=
=
f

g
x
=
f
g
x
=
f
o
g
x
Therefore, resulting “fog” function is even function.
The nature of resulting function subsequent to various mathematical operations is tabulated here for reference :

f(x) g(x) f(x) ± g(x) f(x) g(x) f(x)/g(x), g(x)≠0 fog(x)

odd odd odd even even odd
odd even Neither odd odd even
even even even even even even

We should emphasize here that we need not memorize this table. We can always carry out particular operation and determine whether a particular operation results in even, odd or neither of two function types. We shall work with a division operation here to illustrate the point. Let f(x) and g(x) be even and odd functions respectively. Let h(x) = f(x)/g(x). We now substitute “x” by “x”,
⇒
h

x
=
f

x
g

x
⇒
h

x
=
f

x
g

x
But f(x) is an even function. Hence, f(x) = f(x). Further as g(x) is an odd function, g(x) =  g(x).
⇒
h

x
=
f
x

g
x
=

h
x
⇒
h

x
=
f
x

g
x
=

h
x
Thus, the division, here, results in an odd function.
There is an useful parallel here to remember the results of multiplication and division operations. If we consider even as "plus (+)" and odd as "minus ()", then the resulting function is same as that resulting from multiplication or division of plus and minus numbers. Product of even (plus) and odd (minus) is minus(odd). Product of odd (minus) and odd (minus) is plus (even). Similarly, division of odd (minus) by even (plus) is minus (odd) and so on.
Square of an even or odd function
The square of even or odd function is always an even function.
Properties of derivatives
1: If f(x) is an even differentiable function on R, then f’(x) is an odd function. In other words, if f(x) is an even function, then its first derivative with respect to "x" is an odd function.
2: If f(x) is an odd differentiable function on R, then f’(x) is an even function. In other words, if f(x) is an odd function, then its first derivative with respect to "x" is an even function.