Relations and Functions

📚 Complete Practice Guide: Relations and Functions

Relations and Functions is a fundamental Class 12 Mathematics topic because it provides the language used to describe how elements of one set are connected to elements of another. A relation can pair elements from two sets, while a function is a special relation in which every element of the domain has exactly one image in the codomain. This DigiSadhan worksheet focuses on the definitions, representations and calculations students commonly practise in this chapter. The supplied NCERT link opens the official NCERT textbook portal, which is labelled “Textbooks PDF (I-XII)”.

1. Relations

If A and B are two non-empty sets, a relation from A to B is a subset of the Cartesian product A×B. An ordered pair (a,b) belongs to the relation when a is related to b. Relations can be represented using ordered pairs, arrow diagrams, tables or suitable mathematical descriptions.

R is a relation from A to B if R ⊆ A×B

2. Domain, Codomain and Range

For a relation from A to B, A is the domain and B is the codomain. The range is the set of actual second components that occur in the relation. In a function, every domain element has exactly one image, but several domain elements may have the same image.

Domain = set of inputs   Codomain = declared target set   Range = actual outputs

3. Function

A function f from A to B is a relation in which every element of A is associated with exactly one element of B. We write f:A→B. If a∈A, the corresponding output is denoted f(a). A useful check is to ask whether each input has one and only one output.

f:A→B means every a∈A has exactly one f(a)∈B

4. One-One Function

A function is one-one or injective if distinct elements of the domain have distinct images. Equivalently, f(a)=f(b) implies a=b. For finite sets, an injective mapping cannot assign two different domain elements to the same output.

f(a)=f(b) ⇒ a=b

5. Onto Function

A function is onto or surjective if every element of the codomain is the image of at least one element of the domain. Thus the range equals the codomain.

f is onto ⇔ Range(f)=Codomain(f)

6. Bijective Function

A function that is both one-one and onto is called bijective. Bijective functions are especially important because they possess inverse functions when the domain and codomain are appropriately specified.

Bijective = One-one + Onto

7. Composition of Functions

If f:A→B and g:B→C, the composition of g with f is written g∘f. The output of f becomes the input of g. Composition is not generally commutative, so g∘f and f∘g should never be assumed equal.

(g∘f)(x)=g(f(x))

8. Invertible Functions

A function has an inverse function when it is bijective between the relevant sets. The inverse reverses the input-output correspondence. If y=f(x), then x=f⁻¹(y). In composition form, inverse functions satisfy identity relationships on the appropriate domains.

f⁻¹(f(x))=x    f(f⁻¹(x))=x

9. Algebra of Functions

When functions have compatible domains, new functions can be formed by addition, subtraction, multiplication and division. The quotient requires the denominator function to be non-zero at the points under consideration.

(f+g)(x)=f(x)+g(x)   (f−g)(x)=f(x)−g(x)   (fg)(x)=f(x)g(x)

10. Domain Restrictions

Domain questions are often where small algebraic errors occur. For a rational expression, exclude values that make the denominator zero. For a real square-root expression, the quantity under the root must be non-negative. For composite functions, the output of the inner function must belong to the domain of the outer function.

11. Common Question Types in This Tool

This worksheet includes more than seven categories: relation identification, domain and range, one-one/onto classification, composition, inverse functions, function operations, concepts and formulae, missing values and application-style problems. MCQ options are independently shuffled every time the worksheet is rendered.

12. Difficulty Strategy

Begin with Easy questions on definitions, ordered pairs and direct function evaluation. Medium practice should include domain/range, classification and composition. Hard practice should combine inverse functions, restricted domains and multi-step compositions. Mixed mode is useful for exam-style revision.

13. Quick Formula Revision

R ⊆ A×B

f:A→B: every input has exactly one output

One-one: f(a)=f(b) ⇒ a=b

Onto: Range(f)=Codomain(f)

(g∘f)(x)=g(f(x))

Bijective ⇔ one-one and onto

14. How to Use the Worksheet

Select question count, mode, topic, difficulty and timer, then press Start New Test. Every question has a Check Answer button. Short-answer questions provide a textbox for the answer. Correct answers turn green and incorrect answers turn red. Show Answer stays locked until checking. Progress, score and accuracy update immediately, and a final result appears after every question has been checked. Shuffle Whole Worksheet changes the question order, while Save as PDF and Print remain separate options.

Exam tip: Always distinguish domain, codomain and range. For one-one tests, compare images of different inputs. For onto tests, check every codomain element. For composition, apply the inner function first. For inverse functions, verify the composition identities and pay close attention to restricted domains.