An Introduction
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 an Increasing Function. Similarly, if the graph is going down then it is called a Decreasing Function.
Increasing Decreasing Functions
Consider the following diagram:
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A Function is called 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 go to the right, hence, it is called an Increasing Function.
A Function is called a Decreasing Function if the value of y decreases with the increase in the value of x. As above, on 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).
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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)
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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.
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)
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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)
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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)
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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)
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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:
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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 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.
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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.
Tips to Study Increasing and Decreasing Functions and Monotonicity
After studying the above content, you might have understood the Functions, Increasing Functions, Decreasing Functions and Monotonic Functions with a lot of examples of each type.
Students are advised to practice a lot of questions to test their knowledge and understanding after the conceptual clarity that they've got after studying the study material.
When it comes to a practical subject like Mathematics, the only thing that can help you perform well is ample practice.
Let's understand some of the tips and tricks to crack the best way to prepare for this subject.
Understand your Study Habits
Since every student has a different study habit and pattern, there cannot be one way that would work for everyone. Hence, students are advised to closely observe and monitor their study habits. Another important point that you shall remember is that you don’t have to always match up with others, your friend might sit and study for 3 hours straight whereas you might need a short break after an hour or so. Understand that both the cases are normal and it depends on you on how you have to plan things for yourself.
Prepare a Realistic Schedule
Preparing a schedule that is realistic, time-based and feasible is a skill that students shall know. It can help you master the technique of managing your time and being the most productive at most times. Students often make some unrealistic schedules that cannot be completed and it does nothing other than demotivate you. So, it is advisable to observe your study habits and then, make a proper plan and make sure to stick to it.
Mind Mapping
One more technique that should be used to keep track of your preparation is making a mind map. It is an integral part of the learning process. This helps the students to know and understand where to put effort and how. Students should follow a goal-oriented approach and make sure to complete the targets at all times.
Keep a Track
Keeping a track of topics that you have prepared and the ones which are left is an important habit that the students shall possess. It helps you to keep a check on what further actions should be taken to move ahead. Apart from this, you shall also be patient and trust the process. If you unnecessarily stress yourself, nothing would be achieved and you will only decrease your productivity levels.
Revision Technique
Revision should be done according to your study habits and not according to what others have been following. The two most effective revision techniques that you may surely do before the examinations are 4R's technique which includes the reading of important topics, revision of notes, reviewing of mock tests and re-reading of difficult topics and the other one is the 3R's technique which includes revising, reciting and review. Both of them are equally important and with this, success is guaranteed.
Pace
The pace is an important factor in the learning process. You shall understand that being too stressed or not bothered at all, both of these things are extremely challenging and bad for you, as students. Hence, you are advised to follow a midway approach towards it which means that you shall be able to maintain a good balance.
Practice
Practice is the most vital step as practising makes your preparation more interesting and fulfilling. It shall include three steps: thorough reading of the study material, going through the notes over and over again and attempting a set of questions regularly. Once you're done with it, the results will amaze you!
FAQs on Increasing and Decreasing Functions and Monotonicity
1. What are increasing and decreasing functions in the context of the CBSE Class 12 syllabus?
In the CBSE Class 12 syllabus, an increasing function is a function where the value of f(x) increases or stays the same as the value of x increases. Graphically, this means the curve moves uphill or remains level as you move from left to right. A decreasing function is one where the value of f(x) decreases or stays the same as x increases, meaning the graph moves downhill or is level. This concept is a key part of the 'Application of Derivatives' chapter.
2. How can derivatives be used to determine if a function is increasing or decreasing on an interval?
The First Derivative Test is used to find the intervals of monotonicity. For a function f(x) that is differentiable on an open interval (a, b):
- If f'(x) > 0 for all x in (a, b), then the function is strictly increasing on that interval.
- If f'(x) < 0 for all x in (a, b), then the function is strictly decreasing on that interval.
- If f'(x) = 0 for all x in (a, b), then the function is constant on that interval.
3. What is the precise difference between a 'strictly increasing' function and a 'non-decreasing' function?
The difference lies in how they handle equality. A function is strictly increasing if for any x₁ < x₂, it must be that f(x₁) < f(x₂); the function value must always rise. A non-decreasing function allows for plateaus, meaning if x₁ < x₂, then f(x₁) ≤ f(x₂). This implies the function can remain constant over some parts of an interval but never decreases.
4. What does it mean for a function to be monotonic?
A function is described as monotonic on a given interval if it is either entirely non-increasing or entirely non-decreasing over that whole interval. In simpler terms, a monotonic function is one that never changes its direction of slope; it consistently moves in one direction (up, down, or flat) without switching between them.
5. Can a function be neither increasing nor decreasing on its entire domain? Provide an example.
Yes, many functions change their behaviour. For example, the function f(x) = x² is not monotonic over its entire domain of real numbers. It is decreasing on the interval (-∞, 0) and increasing on the interval (0, ∞). Similarly, trigonometric functions like sin(x) and cos(x) are periodic and have intervals of both increase and decrease.
6. If the derivative f'(c) = 0 at a point 'c', does it guarantee the function is neither increasing nor decreasing at that point?
Not necessarily. A point where f'(c) = 0 is a critical point. This point can be a local maximum or minimum, where the function changes direction. However, it can also be a point of inflection where the function continues its monotonic behaviour. For example, the function f(x) = x³ has a derivative f'(x) = 3x², which is zero at x=0. Despite this, f(x) = x³ is a strictly increasing function across its entire domain, including at x=0.
7. What are some real-world applications or examples of increasing, decreasing, and monotonic functions?
These concepts appear frequently in various fields:
- Increasing Function: The total revenue generated by a company as it sells more units of a product.
- Decreasing Function: The radioactive decay of an unstable isotope over time, or the depreciation in the value of a car.
- Monotonic Function: A person's height from birth to adulthood is a non-decreasing (monotonic) function; it increases or stays the same but does not decrease.
