CBSE Class 12 Maths: Relations and Functions – Comprehensive Study Guide
For CBSE Class 12 students, mathematics is a required course; “Relations and Functions” is among the fundamental subjects taught in the curriculum. Higher mathematical ideas include calculus, algebra, and set theory are laid in this chapter. To help students learn this chapter, this blog will go over key ideas, formulas, solved examples, and pertinent questions from the CBSE Class 12 Maths – Relations and Functions Notes & Question Bank.
Introduction to Relations and Purpose
Simply said, a relation is a relationship between members of two sets; a function is a particular kind of relation whereby every input has a unique output. Solving algebraic and practical problems requires a grasp of these ideas.
In mathematics, relations
A relations between two sets is a subset of the Cartesian product of those sets. It shows the interactions among the elements of two sets.
Variations of Relations:
An empty relation R in a set A is one in which no element of A is connected to any other element of A.
R is an empty relations, for instance, if A = {1, 2, 3} and R = ∅.
A relation R in a set A is universal if every element of A relates to every other element of A.
R is a universal relation, for instance, if A = { 1, 2, 3} and R = A × A.
A reflexive relation R in set A is one whereby (a, a) ∈ R for every a ∈ A.
R is reflexive, for instance, if A = { 1, 2, 3} R = {(1,1), (2,2), (3,3), (1,2)}.
A symmetric relation R in set A is one whereby (a, b) ∈ R implies (b, a) ∈ R.
For instance, if (2,3) ∈ R, then (3,2) has to likewise be in R if it is symmetric.
A relation R in set A is transitive if (a, b) ∈ R and (b, c) ∈ R, so (a, c) ∈ R.
For instance, (2,3) ∈ R and (3,4) ∈ R; so, (2,4) must also be in R if it is transitive.
An equivalency relation is a relation that is reflexive, symmetric, and transitive.
In mathematics, functions
A function is a particular kind of relation whereby every input has a distinct output. Denoted f: A → B, A is the domain and B is the codomain.
Various kinds of purposes:
One-one function, sometimes known as injective function
One-one is the function f: A → B whereby unique items in A have unique images in B.
One-one is f(x) = x + 2, for instance.
2.onto function—surjective function
Every element in B has at least one pre-image in A, hence a function f: A → B is onto.
For instance, if B = R (set of real numbers), f(x) = x² is not onto; but, if B = R⁺ (set of non-negative real numbers), it is onto.
3. Bijective Goal
Bijective is a function with both one-one and onto character. Its function is inverted.
One bijective function is f(x) = x³.
Crucial Characteristics of Functions
1. Range and Domain
Domain: The set of all conceivable A inputs.
Range: Subset of B’s whole range of possible output values.
2. Combining purposes
The composite function (g ∘ f) defines itself by (g ∘ f)(x) = g(f(x)) if f: A → B and g: B → C.
Third: inverse of a function
A function is bijective if its inverse f⁻� such that f(f⁻¹(x)) = x.
Class 12 Important Questions from CBSE
One-Mark inquiries
Determine the range of R by first finding x + 2y = 8′.
Take A = {1,2,3}. Write the smallest equivalent relation on A.
3. Indicate if for x ∈ R the function f(x) = x² is one-one or not.
Two-Mark Assignments
Allow R to be the relation “is greater than” on A from {2,4,6,8}. Write R as a collection of ordered pairs and argue if it is transitive.
Given A = {1,2,3,4} and B = {-1,3}, how many onto functions run from A to B?
Four-mark Questions
Show that f(x) = x² does not have either one-one nor onto effect on N → N.
Show that the equivalency relation R in A = {5,6,7,8,9} defined by R = {(a,b): |a-b| is divisible by 2}.
Six-Mark Examining Questions
Examining f: R → R defined by f(x) = 5x² + 6x – 9, one finds Show that f lacks invertibility.
Inverse of the function f(x) = x³ + 4.
Strategies to Perfect Relations and Functions for CBSE Class 12
Solving issues depends on knowing the definitions of several kinds of relations and functions.
Solve several kinds of problems, particularly on equivalency relations, injective and surjective functions, using your practice conceptual questions.
Graphs and diagrams provide a visual means of illustrating relations and functions, therefore facilitating understanding.
4. Memorize Key Formulas: Save a formula sheet close at hand for fast review.
Using CBSE past year exam papers would allow one to grasp the pattern of questions.
In summary
Higher mathematics rests on relations and functions. Learning these ideas can benefit you not only on CBSE board tests but also on other entrance tests including JEE and other competitive ones. Students who practice consistently and have a good awareness of qualities will find it simple to score well in this chapter.
Maintain your practice; all the better for your CBSE Class 12 Maths tests!
