Decreasing Functions

Introduction

A function refers to the relation between input and output in mathematics. It can either be increasing, decreasing or even constant. That happens for the given intervals throughout a particular domain. Moreover, they are differentiable and continuous at the given interval. Talking about an interval, it is the connection or the continuous portion or part of a real line. The increasing or decreasing functions are mostly used in the application of derivatives. So, if you want to know whether a function is increasing or decreasing, you can easily calculate the same using derivatives. The properties, theorems, and graphical representations of these functions are easy to understand.

Application of Derivatives

Functions can either increase, decrease, or remain constant throughout their respective domains. Now, both increasing and decreasing functions are one of the most used applications of derivatives. The latter is to identify whether the given function increases or decreases at a given interval. If something moves upwards, it is increasing. If it moves downwards, then it decreases. Speaking graphically, if the graph of the given function goes upwards, then it is an increasing function. Similarly, if the graph of the same function goes downwards, then it is a decreasing function.

Monotonic Functions

The increasing or decreasing behaviour of functions is termed their monotonicity. In simple words, the monotonic function is the one between ordered sets that either preserves or reverses a given order. It has two terms included – the first is mono which refers to one, and the other one is tonic which refers to the tone.

Decreasing Functions

Decreasing functions are the ones whose graphs go downwards as you move towards the right-hand side of the x-axis. These are also called non-increasing functions. These functions decrease at a given interval. In other words, when a function or f(x) is decreasing, the values of the same are also decreasing with an increase in x. From the derivative point of view, a function decreases when a derivative at a particular point is negative. Decreasing functions include intervals where the function decreases and where the same is constant. 

Properties of Decreasing Functions

Now that we know how to check whether a function is increased or decreased, let us go through the algebraic properties of decreasing functions:

1. If the functions titled f and g are decreasing functions on an open interval the sum of the functions will also decrease on this interval.
2. If the function titled f is a decreasing function on an open interval then its opposite function -f will increase on this interval.
3. If the function titled f is a decreasing function on an open interval or I – then the inverse function or 1/f will increase on this interval.
4. If the functions titled f and g are decreasing functions on an open interval or I and f, g ≥ 0 on I – the product of the functions or fg will also decrease on this interval.

Steps to Calculate Decreasing Functions

Following are the steps involved in calculating decreasing functions:

1. The given function is always differentiated with respect to the given constant variable.
2. The first derivative is always in the form of an equation. That is to provide the value of x at which derivative =0. So, f(x) = 0.
3. We get the open intervals along with the value of x. That is complete after solving the equation of the first derivative and the points of discontinuity. Through the same process, you must consider the sign of the intervals.
4. The points of the interval that are formed are compared to find the sign in the first derivative. If the sign of the interval in the first derivative form displays a value more than 0, then the function is increasing in nature. On the other hand, if the sign of the interval displays a value less than 0, then the function is decreasing in nature.
5. Finally, we get the increasing or decreasing intervals of the said function.

The nature of the function can be identified through a specific test. It is called the first derivative test.

Identification of Decreasing Functions

Identifying whether a function is decreasing or not is very easy. First, you need to take a derivative and then set it equal to zero. After that, you must find between which zeros are the functions negative. Next, you need to test values on all sides of these derivatives to find when the said function is negative or decreasing. That is how you can identify whether any given function is increasing or decreasing.

Important Applications of Functions

The important applications of both increasing and decreasing functions that are used in real life are listed below:

1. Functions are used in the mathematical building blocks meant for designs.
2. These are utilised in machines that predict disasters.
3. They are also useful in curing diseases.
4. Functions are further used to understand the world economy.
5. They help keep aeroplanes in the air.

Mathematical Applications of Functions

1. Algebra
2. Fourier analysis and differential equation
3. Computational and differential geometry
4. Statistics and probability
5. Numerical analysis
6. Optimisation and operational research

Conclusion

Functions are an integral part of mathematical calculations. Whether increasing, decreasing or constant, these are applied in various applications. The above study material notes on decreasing functions explain its definition, properties, identification, and application. Decreasing functions are among the most used applications of derivatives. They have substantial differences as compared to increasing functions. You can identify whether a function is increasing or decreasing through the process of differentiation. You need to find the derivative of the given function and determine the critical point.