The properties of a function, particularly its behavior over its domain, can be used to classify it. A monotonic function is a function that is either entirely non-increasing or non-decreasing.

Non-Increasing Function

A non-increasing function either stays the same or decreases as its input increases. Algebraically, a function $f(x)$ is non-increasing if, for any two points $x_1$ and $x_2$ such that $x_1\le x_2,$ the corresponding function value $f(x_1)$ is at least $f(x_2).$

On a graph of a non-increasing function, regardless of the starting point, moving right keeps the output value the same or makes it lower. On a related note, a decreasing function always makes its output smaller as the input gets larger. Since every decreasing function fulfills all the requirements of a non-increasing function, it can be viewed as a special case of it.

Non-Decreasing Function

A non-decreasing function either stays the same or increases as its input increases. In algebraic terms, a function $f(x)$ is non-decreasing if for any two points $x_1$ and $x_2$ such that $x_1\le x_2,$ the corresponding function value $f(x_1)$ is at most $f(x_2).$

In a graph of a non-decreasing function, moving right keeps the output value the same or makes it greater. Note that an increasing function always makes its output bigger as the input gets larger. Since every increasing function fits the rules of a non-decreasing function, it is a special case of it.