Introduction

To understand monotonicity (increasing and decreasing) of functions, it is essential to understand what functions are. In simple words, functions can be understood as a relationship between the input and output so that every input is related to one and only one output.

You will find increasing, decreasing, and even constant functions. Functions are continuous as well as differentiable in a given interval. Let us develop a proper understanding of what functions are, and then we shall move on to monotonic functions. These functions hold a lot of importance in mathematics at the school level as well as for higher studies, and students should study the topics properly.

What are functions?

Functions are a relationship between inputs and outputs. In a function, every input is related to one and only one output. Functions have a domain, range, and codomain. f(x) is used to denote functions where x is the input. Functions can also be represented as y = f(x). In mathematics, the set of inputs is  known as the domain, and the set of outputs is known as range, which is a subset of codomain. There are several types of functions such as:

• Surjective functions: They are also known as onto functions when more than one element is mapped from the domain with one element of range
• Injective functions: They are also known as one-to-one functions
• Inverse functions: Functions which are one-one and onto, are invertible
• Polynomial functions: Those functions that have polynomials are known as polynomial functions

Let us now discuss monotonic functions.

Monotonicity of a function

The monotonicity of a function can be described as the increasing or decreasing nature of a function. Monotonic functions are those functions that follow any one of these cases:

• If x and X  are two endpoints in an interval and x< X then
•  If f(X) is less than or equal to f(x), then the function is a decreasing function
• If f(X) is greater than or equal to f(x), the function is known as an increasing function
• If f(X) is always greater than f(x), the function is known as strictly increasing
• If f(X) is less than f(x), the function is known as strictly decreasing

We will now look at increasing and decreasing functions closely

Increasing functions

Any function that is increasing at a given interval is known as an increasing function. I suppose there are two endpoints of an interval, namely, x and X, where X>x. In the case of these functions, f(x) Is less than or equal to f(X), which is quite different from a strictly increasing function. This inequality makes the function unique in the sense that some parts of this function are similar to both functions, which makes it different from strictly increasing functions. A strictly increasing function is the one where f(x) is less than f(X); it can also be written as f (x) < f (X). 

Decreasing functions

Any function which decreases in a given interval is known as a decreasing function. Now suppose that there are two endpoints in an interval, namely x and X, and f (x) is greater than or equal to f(X), then the function is a decreasing function. If however, f(x) is greater than f(X) i.e f(x) > f(X) then the function is strictly decreasing. 

Monotonic functions

• Monotonic functions are those functions that can be differentiated in a given interval of time and that are included in any one of the following categories:
• Increasing function
• Strictly increasing function
• Decreasing function
• Strictly decreasing function 

A function where df/dx = 0 is constant in a given interval.

Monotonic functions can be further explained with the help of the first derivative test, which is discussed below.

 First derivative test

Let us now understand monotonic functions with the help of the first derivative test. The derivative shows the behaviour of a function at different values.

For all the values of x, if df/dx ≤ 0, then the function is a decreasing function 

For all the values of x, if df/dx ≥ 0, then the function is an increasing function

Strict equalities can be shown as follows:

The function will be strictly increasing if df/dx > 0 

The function will be strictly decreasing if df/dx < 0

Test for the increasing and decreasing functions

To test for the increasing and decreasing functions, we will use derivatives of a function. For all the four cases that we had discussed earlier (i.e., Increasing function, strictly increasing function, decreasing function, and strictly decreasing function), we have tests. However, there is something to remember before we discuss these tests. To test the monotonicity of a function f, we first calculate its derivative f′. The function should be continuous in the [a,b] interval, and it should be differentiable in (a,b). The tests are given as follows:

For a non-increasing function, the test is: f’ (x) ≤0, ∀ x ∈ (a,b)

For a non-decreasing function, the test is: f’ (x) ≥ 0, ∀ x ∈ (a,b)

For a decreasing and strictly decreasing function the test is:

f’(x) < 0 ∀ x ∈ (a,b)

For an increasing and strictly increasing function, the test would be as follows:

f’(x) > 0 ∀ x ∈ (a,b)

The monotonicity of a function can be described as the increasing or decreasing nature of a function. Suppose we have given two points x₂ < x₁, then increasing functions are the ones where f[x₁] is greater than or equal to f[x₂] and a decreasing function is ones where f[x₁] is lesser than or equal to f[x₂], a strictly increasing function is one where f[x₁] is greater than f(x₂) and a strictly decreasing function is one where f[x₁] is lesser than f[x₂].