# Nature of a Function (In Terms of Equation)

On the basis of the symmetric nature of functions, we define the graphs to be of three natures:

# Even Function

For y = f(x),

Mathematically, even functions are defined as y = f(x) = f(âˆ’x)

In layman terms, if we replace â€˜xâ€™ from the given equation by â€˜â€“xâ€™, then the equation should be the same.

For example, y = f(x) = x2 + 8.

Here, f(âˆ’x) = (âˆ’x)2 + 8 = x2 + 8 = f(x)

So, y = f(x) = x2 + 8 is an even function.

Properties of even function
1. The sum or the difference or the product or the division of any two even functions is an even function.
2. For y = f(x), the graph of the even functions are symmetrical to Y-axis.

# Odd Function

For y = f(x),

Mathematically odd functions are defined as y = f(x) = âˆ’ f(âˆ’x)

In layman terms, if we replace â€˜xâ€™ from the given equation by â€˜â€“xâ€™, then the equation should be the same with an opposite sign.

For example, y = f(x) = x3.

Here f(âˆ’x) = (âˆ’x)3 = âˆ’(x3) = âˆ’f(x)

So, y = f(x) = x3 is an odd function.

Properties of odd function
1. The sum or the difference of any two odd functions is an odd function.
2. For y = f(x), the graph of the odd functions are symmetrical to the origin.

# Neither Odd Nor Even Function

For y = f(x),

If any equation is not satisfying the conditions of either an odd or an even function, then it is said to be neither odd nor even function.

In layman terms, if we replace â€˜xâ€™ from the given equation by â€˜â€“xâ€™, then the equation should not be the same, either with the same sign or with an opposite sign.

For example, y = f(x) = x3 + 1

Here, f(âˆ’x) = (âˆ’x)3 + 1 = âˆ’x3 + 1

It is quite clear now that f(âˆ’x) is neither equal to f(x) or â€“f(x).

So, y = f(x) = x3 +1 is neither an even function nor an odd function.

Properties of neither odd nor even function
1. The sum of the difference of odd and even functions is neither an odd nor an even function.
2. To understand the mechanism of the sum/difference/product of two functions, we can take an even function as a positive number and an odd function as a negative number.

For example, x2 (Even function) Ã— x3 (Odd function) = Odd function

x2 (Even function) + x3 (Odd function) = Neither odd nor even function

Remember that no function can be both an even and an odd function simultaneously.
Example
Find the nature of the following functions:
1. y = f(x) = e-x
2. y = f(x) = x2 + x4
3. y = f(x) =
4. y = f(x) = x2 + x3
5. y = f(x) = logx2
6. y = f(x) = logax
Solution
1. Neither odd nor even
2. Even
3. Odd
4. Neither odd nor even
5. Even
6. Neither odd nor even