In algebra, function can be classified by their nature. Just like even and odd numbers, we have even and odd functions in algebra. An even and odd function are connect to symmetric of the given function. The symmetric of a function is visible by the graph of that function. Graph can be symmetric about x-axis or y-axis or origin.
Sometimes, function is not symmetric, they may neither be odd nor be even. To know about a function is even or odd is determined by some algebraic algorithm.
Now we discuss more about even and odd functions.

Even Functions

Let f : X$\rightarrow$ Y be a given function, then f is said to be an even function if,
f(x) = f(-x), for every x$\in$ X, i.e. if we replace x by -x then the function remains same.
f(x) = x2, f(x) = x4, f(x) = cos x etc.. are even function.

If we plot a graph of an even function then, it is symmetric about Y-axis.

Odd Functions

A function f : X$\rightarrow$ Y is called an odd function if,
f(-x) = -f(x), $\forall$ x$\in$X.

Like f(x) = x, f(x) = x3, f(x) = sin x, etc... are an odd function. And graph of an odd function is symmetric about the origin.

Even and Odd Functions Examples

Example 1:
Show that f(x) = x2 cos x is an even function.
Solution:
Given that f(x) = x2 cos x
Replace x by -x then,
f(-x) = (-x)2 cos (-x)
= x2 cos x
= f(x)
Hence given function is an even function.

Example 2:
Prove that f(x) = x sin x is an even function.
Solution:
Replace x by -x in the given function then, we get
f(-x) = (-x) sin (-x)
= (-x) (-sin x)
= x sin x
= f(x)
Given function is an even.

Example 3:
Show that f(x) = x cos x is an odd function.
Solution:
Given that f(x) = x cos x, replace x by -x then,
f(-x) = (-x) cos (-x)
= -x cos x
= -f(x)
So f is an odd function.

Example 4:
Show that f(x) = cosh x is an even function.
Solution:
Since we know that cosh x = $\frac{e^{x}+ e^{-x}}{2}$.
Now we replace x by -x then
f(-x) = cosh (-x) = $\frac{e^{-x}+ e^{-(-x)}}{2}$
= $\frac{e^{-x}+ e^{x}}{2}$
= cosh x
= f(x)
So f is an even function.
Note:
• If we multiply an even function with an odd function, then we get an odd function, so multiplication of even odd is odd.
• If we multiply an even function with an even function, then we get an even function so multiplication of even even is even.
• If we multiply an odd function with an odd function, then we get an even function so multiplication of odd odd is odd.
• If we multiply an odd function with an even function, then we get an odd function so multiplication of odd even is odd.

Even Odd or Neither Functions

Let f(x) = x3 - 2 x2 + 4 x + 1 be a function and we will test whether it is even or odd.
To calculate f(-x), we replace x by -x then
f(-x) = (-x)3 - 2 (-x)2 + 4 (-x) + 1,
or f(-x) =- x3 - 2 x2 - 4 x + 1.

This function is not equal to the given function f(x) and not the opposite of that.
" If a function is not equal to f(x) or f(-x) {i.e. opposite of f(x) }, then such type of function is said to be neither evevn nor odd".