 # Increasing And Decreasing Functions And Monotonicity

In this article, we would be discussing the increasing and decreasing functions. But before we proceed with it, let us discuss what a function is. A function is called a relation between the input and the output in a way that each input is related to exactly one output.

Functions can either increase, decrease or remain constant for intervals throughout their entire domain. They are continuous and differentiable in the intervals given. An interval defined as a continuous or connected portion on the real line.

Increasing decreasing function is one of the most used applications of derivatives. Derivatives are used for identifying if the given function is increasing or decreasing in a specific interval.

You would know that if something is increasing, it is moving upwards and if something is decreasing it is moving downwards. Therefore, if you talk graphically if the graph of the function is going upwards, then it is called as an increasing function. Similarly, if the graph is going down then it is called a decreasing function.

### Increasing Decreasing Functions

Consider the following diagram:

A function is called as an increasing function if the value of y increases with the increase in the value of x. As you can see from the above figure that at the right of the origin, the curve is moving upwards as you goto the right, hence, it is called an increasing function.

A function is called as a decreasing function if the value of y decreases with the increase in the value of x. As above, in the left of the origin, the curve is moving downwards if you move from left to right.

### Increasing Function Definition

Here is the definition of a function that is increasing on an interval.

Consider a function y = f(x)

The function is increasing over an interval, if for each x1 and x2 in the interval, x1 < x2, and f( x1) ≤ f(x2).

It is a strictly increasing function over an interval, if for each x1 and x2 in the interval, x1 < x2, and

f( x1) < f(x2)

You can see that there is a difference in the symbols in both the above increasing functions.

### Decreasing Function Definition

Consider a function y = f(x)

This function is decreasing over an interval , if for each x1 and x2 in the interval, x1 < x2, and

f( x1) ≥ f(x2)

A function is a strictly decreasing function over an interval, if for each x1 and x2 in the interval, x1 < x2, and f( x1) > f(x2).

You can notice that there is a difference in the symbols in both the above decreasing functions.

Monotonic Functions

The increasing or decreasing behaviour of the functions is referred to as monotonicity of the function.

A monotonic function is referred to as any given function that follows one of the four cases mentioned above. Monotonic generally has two terms in it. Mono refers to one and tonic refers to tone. Both these words together mean “in one tone”. When you say that a function is non-decreasing, does it mean that it is increasing? The answer is no. It can also mean that the function does not vary at all. In simpler words, the function is having a constant value for a particular interval. Make sure to not confuse non-decreasing with increasing.

Increasing And Decreasing Functions Examples

Now, let us take a look at the example of increasing function and decreasing function. The concepts that are explained above about the increasing functions and the decreasing functions can be represented in a more compact form.

1. Increasing Or Non-Decreasing

A function y = f(x) is called increasing or non-decreasing function on the interval (a, b) if

∀ x1, x2 ∈ (a, b): x1 < x2

f (x1) ≤ f(x2)

1. Strictly Increasing

A function y = f(x) is called strictly increasing function on the interval (a, b) if:

∀ x1, x2 ∈ (a, b): x1 < x2

⇒f(x1) < f(x2)

1. Decreasing Or Non-Increasing Function

A function y = f(x) is called decreasing or non-increasing function on the interval (a, b) if:

∀ x1, x2  ∈(a, b): x1 < x2

⇒f(x1) ≥ f(x2)

1. Strictly Decreasing Function

A function y = f(x) is called strictly decreasing function on the interval (a, b) if:

∀ x1, x2 ∈ (a, b): x1 <x2

⇒f(x1) > f(x2)

If the given function f(x) is differentiable on the interval (a,b) and belongs to any one of the four considered types, that is, it is either increasing, strictly increasing, decreasing, or strictly decreasing, the function is called monotonic function on this particular interval.

Monotonic Function

Monotonically Increasing Functions

The graphs of both the exponential and the logarithmic functions are important. From these graphs you can see a general rule:

If a>1, then both of these functions are monotonically increasing:

f(x)=ax

g(x)=loga(x)

Monotonically Increasing Function Example

Consider the given two graphs:

The red graph is denoted by f(x) = 3x and the green graph is denoted by g(x) = 3x+1.

When x is increasing, f(x) is also increasing. Hence,

g(x) = 3x+1

= 3 . 3x

= 3 f(x)

Hence, g(x) is a monotonically increasing function.

### Monotonically Decreasing Function

A monotonically decreasing function is basically the opposite of monotonically increasing functions.

If f(x) is a monotonically increasing function over a given interval, then −f(x) is a said to be a monotonically decreasing function over that same interval, and vice-versa.

### Monotonically Decreasing Function Example

Consider the following graph where f(x) = -5x.

As you can see that the function 5x is monotonically increasing here, hence, f(x) = -5x should be monotonically decreasing.

In the graph, when 5x increases, f(x) decreases.