TN Board Class 10 Maths Chapter 1: Relations and Functions

About Relations and Functions

Medium ~120 min study

The concept of sets forms the baseline of modern mathematical language. In this chapter, we build upon set theory to establish connections between elements of different sets. These connections, known as relations, model a wide range of real-world pairings, from seat assignments in an auditorium to physical measurements of objects. Understanding how mathematical objects relate to one another allows us to structure data and define systematic patterns.

Functions represent a special class of relations where every input is paired with exactly one unique output. This critical concept acts as a mathematical machine, translating one set of variables into another. By studying functions, students learn to model situations such as speed as a function of time, or price as a function of demand. This chapter categorizes these mappings into several types, providing the necessary toolkit to analyze structured algebraic relationships.

From an examination perspective, this chapter is crucial as it lays the groundwork for calculus, advanced algebra, and coordinate geometry. Questions frequently test the ability to check if a relation is a function, determine domain and range, or find composite functions like \(f(g(x))\). Mastery of these topics ensures a strong start in high school mathematics, helping students tackle both conceptual multiple-choice questions and long-form analytical problems.

What you'll learn

• Define and calculate the Cartesian product between any two sets.
• Identify relations as specific subsets of Cartesian products.
• Distinguish functions from general relations using unique mapping criteria.
• Represent functions using tables, arrow diagrams, ordered pairs, and graphs.
• Classify functions as one-to-one, many-to-one, onto, or into.
• Compute the composition of two or three given functions.
• Recognize linear, quadratic, cubic, and reciprocal functions from their graphs.

Before you start

• Basic set theory including set notations, union, and intersection.
• Familiarity with coordinate systems and plotting points on a Cartesian plane.
• Understanding of basic algebraic equations and solving linear relationships.

Relations and Functions explained

Chapter Overview

Cartesian Product of Sets

The chapter begins by extending set operations to Cartesian products. Given two non-empty sets A and B, their Cartesian product is the set of all ordered pairs (a, b) such that the first coordinate belongs to A and the second belongs to B. This operation represents all possible pairings between elements of the sets. The method is foundational for defining geometric coordinates on a plane, where the entire real plane is viewed as the Cartesian product of the set of real numbers with itself.

Understanding Relations

A relation is mathematically defined as a subset of the Cartesian product. If we have sets A and B, any relationship established between their elements via a specific rule forms a relation. We represent these connections using ordered pairs, arrow diagrams, or set-builder notation. The domain of a relation consists of all first coordinates in these pairs, the co-domain is the entire target set, and the range contains all second coordinates that are actually paired with an element from the domain.

The Concept of Functions

A function is a special type of relation where every element in the domain has a unique image in the co-domain. This means no single input can map to multiple outputs. Functions can be represented in multiple formats: as a set of ordered pairs, a table, an arrow diagram, or a graph. Students can verify whether a graph represents a function by applying the vertical line test, which states that a vertical line must intersect a function's curve at most once.

Classifying Types of Functions

Functions are classified based on how elements in the domain map to the co-domain. A one-to-one function, or injection, maps distinct domain elements to distinct co-domain elements. A many-to-one function maps multiple inputs to the same output. An onto function, or surjection, has a range equal to its co-domain, meaning every target element has a pre-image. If a function is both one-to-one and onto, it is a bijection, representing a perfect one-to-one correspondence between sets.

Composition of Functions

Composition of functions is the process of chaining operations where the output of one function becomes the input of another. For two functions f and g, the composite function is represented as f(g(x)). This operation is generally non-commutative, meaning f(g(x)) does not equal g(f(x)) in most cases. However, function composition is associative, meaning that when three functions are composed, the grouping of the operations does not change the final resulting function.

Graphical Identification of Functions

Finally, the chapter covers how to identify specific algebraic functions using their geometric graphs. Students analyze the characteristic shapes of linear functions, which form straight lines, and quadratic functions, which generate parabolic curves. It also describes cubic curves and reciprocal graphs, where the function is undefined at zero. Recognizing these visual patterns helps students quickly link algebraic equations with their geometric counterparts in higher coordinate mathematics.

Common mistakes to avoid

• Confusing the Cartesian product order: remember that set A cross B is not equal to set B cross A unless the sets are identical.
• Identifying relations that are not functions: always verify that every input element maps to exactly one, and not multiple, output elements.
• Assuming function composition is commutative: keep in mind that composition of f and g is generally not equal to composition of g and f.
• Misidentifying onto functions: check if the range is equal to the co-domain, as any unmapped co-domain element makes it an into function.
• Applying the horizontal line test incorrectly: use the vertical line test for testing functions, and the horizontal line test only to check if a function is one-to-one.

Test yourself on these with the practice test, then check the worked reasoning in the solved MCQs.

Frequently asked questions

What is the difference between a relation and a function?

A relation is any set of ordered pairs representing a connection between two sets. A function is a stricter type of relation where each element in the input domain is associated with exactly one unique element in the output co-domain. Thus, while every function is a relation, not every relation qualifies as a function.

How does the vertical line test work?

To perform the vertical line test, look at the graph of a curve and imagine drawing vertical lines through it. If any vertical line intersects the curve at more than one point, the graph does not represent a function. If every vertical line crosses at most once, it is a function.

What is a bijective function?

A bijective function, or bijection, is a mapping that is both one-to-one and onto. This means that every element in the domain has a distinct image in the co-domain, and every element in the co-domain has a pre-image in the domain, establishing a perfect pairing between the two sets.

Why is function composition not commutative?

Function composition is not commutative because the order of operations matters. Applying one function first and then another changes the intermediate value that the second function processes. For example, squaring a number and then adding three gives a different result than adding three and then squaring the sum.

What are the domain and range of a function?

The domain is the complete set of all possible input values for which the function is defined. The range is the set of all actual output values that the function produces when the inputs from the domain are processed. The range is always a subset of the co-domain.

How do you find the Cartesian product of two sets?

To find the Cartesian product, pair each element of the first set with every element of the second set to create ordered pairs. If the first set has three elements and the second has two, the Cartesian product will contain six distinct ordered pairs containing all combinations.

What is a null relation?

A null relation is a relation from one set to another that contains no ordered pairs. This happens when the defining rule of the relation is impossible to satisfy for any elements in the sets, resulting in an empty set of pairs.