Chapter 5

Chapter 5: Relations and Functions

Chapter Overview

Mathematics is a subject that deals with numbers, quantities, and shapes. It is a vital part of our daily lives and is used in various fields such as science, technology, engineering, and mathematics (STEM). In this chapter, we will explore the concept of relations and functions, which are fundamental in mathematics. Relations and functions are used to describe the relationships between variables and are crucial in understanding many mathematical concepts.

Learning Objectives

• Define and identify relations and functions.
• Understand the difference between a relation and a function.
• Learn to represent relations and functions graphically.
• Understand the concept of domain and range of a function.
• Learn to find the domain and range of a function.
• Understand the concept of composition of functions.
• Analyze and solve problems involving relations and functions.

Important Concepts

Relations

A relation is a set of ordered pairs. It is a way of representing the relationship between two variables. Relations can be represented graphically using a Cartesian plane. For example, consider the relation R = {(1, 2), (2, 3), (3, 4), (4, 5)}.

| x | y |
| --- | --- |
| 1  | 2 |
| 2  | 3 |
| 3  | 4 |
| 4  | 5 |

This relation represents the relationship between two variables x and y. The ordered pairs (1, 2), (2, 3), (3, 4), and (4, 5) indicate that when x = 1, y = 2, when x = 2, y = 3, and so on.

Functions

A function is a special type of relation where each input has a unique output. In other words, for every input, there is only one output. Functions can be represented graphically using a Cartesian plane. For example, consider the function f(x) = 2x.

| x | f(x) |
| --- | --- |
| 1  | 2    |
| 2  | 4    |
| 3  | 6    |
| 4  | 8    |

This function represents the relationship between the input x and the output f(x). The ordered pairs (1, 2), (2, 4), (3, 6), and (4, 8) indicate that when x = 1, f(x) = 2, when x = 2, f(x) = 4, and so on.

Types of Functions

One-to-One Function

A one-to-one function is a function where each output corresponds to only one input. In other words, if f(a) = f(b), then a = b. For example, consider the function f(x) = 2x + 1.

| x | f(x) |
| --- | --- |
| 1  | 3    |
| 2  | 5    |
| 3  | 7    |
| 4  | 9    |

This function is one-to-one because each output corresponds to only one input. If f(a) = f(b), then a = b.

Onto Function

An onto function is a function where every element in the range is the image of some element in the domain. In other words, for every y in the range, there is an x in the domain such that f(x) = y. For example, consider the function f(x) = x^2.

| x | f(x) |
| --- | --- |
| 1  | 1    |
| 2  | 4    |
| 3  | 9    |
| 4  | 16   |

This function is onto because every element in the range is the image of some element in the domain.

One-to-Many Function

A one-to-many function is a function where each output corresponds to more than one input. In other words, if f(a) = f(b), then a ≠ b. For example, consider the function f(x) = x^3.

| x | f(x) |
| --- | --- |
| 1  | 1    |
| 2  | 8    |
| 3  | 27   |
| 4  | 64   |

This function is one-to-many because each output corresponds to more than one input.

Many-to-One Function

A many-to-one function is a function where more than one input corresponds to the same output. In other words, if f(a) = f(b), then a ≠ b. For example, consider the function f(x) = |x|.

| x | f(x) |
| --- | --- |
| 1  | 1    |
| -1 | 1    |
| 2  | 2    |
| -2 | 2    |

This function is many-to-one because more than one input corresponds to the same output.

Domain and Range

The domain of a function is the set of all possible input values. The range of a function is the set of all possible output values. For example, consider the function f(x) = 2x.

| x | f(x) |
| --- | --- |
| 1  | 2    |
| 2  | 4    |
| 3  | 6    |
| 4  | 8    |

The domain of this function is {1, 2, 3, 4} and the range is {2, 4, 6, 8}.

Composition of Functions

The composition of two functions is a new function formed by combining the two functions. For example, consider the functions f(x) = 2x and g(x) = x^2.

| x | f(x) | g(x) |
| --- | --- | --- |
| 1  | 2    | 1    |
| 2  | 4    | 4    |
| 3  | 6    | 9    |
| 4  | 8    | 16   |

The composition of f and g is (f ∘ g)(x) = f(g(x)) = f(x^2) = 2x^2.

Advanced Sections

Deep-Dive Case Studies and Real-Life Applications

Relations and functions are used in various fields such as science, technology, engineering, and mathematics (STEM). For example, the relationship between the speed of a car and the distance it travels is a function. If we know the speed of the car, we can calculate the distance it travels using the formula d = rt, where d is the distance, r is the speed, and t is the time.

Step-by-Step Problem Solving Strategies & Detailed Proofs

To solve problems involving relations and functions, we can use the following step-by-step strategy:

1. Read the problem carefully and understand what is being asked.
2. Identify the variables and the relationships between them.
3. Determine the type of function or relation being described.
4. Use the definition of the function or relation to solve the problem.
5. Check the solution by plugging it back into the original equation.

For example, consider the problem: If f(x) = 2x + 1 and g(x) = x^2, find (f ∘ g)(x).

1. Read the problem carefully and understand what is being asked.
2. Identify the variables and the relationships between them.
3. Determine the type of function or relation being described.
4. Use the definition of the function or relation to solve the problem.
5. Check the solution by plugging it back into the original equation.

Higher-Order Thinking Skills (HOTS) Questions

1. If f(x) = 2x + 1 and g(x) = x^2, find (f ∘ g)(x).
2. If f(x) = x^2 and g(x) = 2x + 1, find (f ∘ g)(x).
3. If f(x) = 2x + 1 and g(x) = x^2, find (g ∘ f)(x).

Previous Year Questions (PYQs) with solutions

1. If f(x) = 2x + 1 and g(x) = x^2, find (f ∘ g)(x). Solution: (f ∘ g)(x) = f(g(x)) = f(x^2) = 2(x^2) + 1 = 2x^2 + 1.
2. If f(x) = x^2 and g(x) = 2x + 1, find (f ∘ g)(x). Solution: (f ∘ g)(x) = f(g(x)) = f(2x + 1) = (2x + 1)^2 = 4x^2 + 4x + 1.
3. If f(x) = 2x + 1 and g(x) = x^2, find (g ∘ f)(x). Solution: (g ∘ f)(x) = g(f(x)) = g(2x + 1) = (2x + 1)^2 = 4x^2 + 4x + 1.

NCERT Textbook Questions & Detailed Answers

Exercise 5.1

1. If f(x) = 2x + 1 and g(x) = x^2, find (f ∘ g)(x). Solution: (f ∘ g)(x) = f(g(x)) = f(x^2) = 2(x^2) + 1 = 2x^2 + 1.
2. If f(x) = x^2 and g(x) = 2x + 1, find (f ∘ g)(x). Solution: (f ∘ g)(x) = f(g(x)) = f(2x + 1) = (2x + 1)^2 = 4x^2 + 4x + 1.
3. If f(x) = 2x + 1 and g(x) = x^2, find (g ∘ f)(x). Solution: (g ∘ f)(x) = g(f(x)) = g(2x + 1) = (2x + 1)^2 = 4x^2 + 4x + 1.

Exercise 5.2

1. If f(x) = 2x + 1 and g(x) = x^2, find (f ∘ g)(x). Solution: (f ∘ g)(x) = f(g(x)) = f(x^2) = 2(x^2) + 1 = 2x^2 + 1.
2. If f(x) = x^2 and g(x) = 2x + 1, find (f ∘ g)(x). Solution: (f ∘ g)(x) = f(g(x)) = f(2x + 1) = (2x + 1)^2 = 4x^2 + 4x + 1.
3. If f(x) = 2x + 1 and g(x) = x^2, find (g ∘ f)(x). Solution: (g ∘ f)(x) = g(f(x)) = g(2x + 1) = (2x + 1)^2 = 4x^2 + 4x + 1.

Exercise 5.3

1. If f(x) =