## Even, Odd, Neither.

The last section of function!!

So, this is the last section of algebra’s function – Even, Odd, and Neither.

## Description:

In algebra, a function is said to be even, odd, or neither based on its symmetry around the y-axis.

Even function:

A function f(x) is even if it satisfies the condition f(-x) = f(x) for all x in the domain of the function. This means that if we reflect the graph of an even function about the y-axis, it remains unchanged. Another way to think about it is that an even function is symmetric with respect to the y-axis. Examples of even functions include f(x) = x^2 and f(x) = cos(x).

Odd function:

A function f(x) is odd if it satisfies the condition f(-x) = -f(x) for all x in the domain of the function. This means that if we reflect the graph of an odd function about the origin, it remains unchanged. Another way to think about it is that an odd function is symmetric with respect to the origin. Examples of odd functions include f(x) = x^3 and f(x) = sin(x).

Neither even nor odd:

If a function does not satisfy either of the above conditions, it is said to be neither even nor odd. For example, the function f(x) = x + 1 is neither even nor odd, since it does not have any symmetry with respect to the y-axis or the origin.

Knowing whether a function is even, odd, or neither can be useful in solving algebraic equations and simplifying expressions, as it allows us to use the properties of these functions to our advantage. For example, if a function is even, we know that its derivative is odd, and vice versa. Similarly, if we have an integral of an odd function over a symmetric interval, we know that the integral is zero.

## Actual handwriting samples:

Probably you don’t need an explanation any further so let’s take a look at some samples.