Sets, Relations and Functions — JEE Main & Advanced Notes

Build domain-range discipline, relation properties, composition and inverse-function thinking before moving into calculus.

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1. Introduction & Exam Weightage

Build domain-range discipline, relation properties, composition and inverse-function thinking before moving into calculus.

Material signal: Mapped from repeated function and inverse-trigonometry revision assignments.

Priority: Must Do. Treat this as a foundation chapter inside the JEE Mathematics ladder.

2. Core Concepts & Definitions

Domain is the input safety check; range is the output possibility set.
A function is one-one when no two allowed inputs share the same output.
Composition requires the range of the inner function to lie inside the domain of the outer function.

3. Key Formulas with Derivation Hints

(f ∘ g)(x)=f(g(x))
Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.
f^{-1}(f(x))=x when f is one-one on the chosen domain
Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.
Number of relations from A to B = 2^{|A||B|}
Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.

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4. Solved Examples — Original Practice Models

Functions: main-style warm-up

Problem: A direct one-step Functions question is solved by identifying the governing definition before substituting values.

Method: write the target quantity, list the known values, use the standard relation, and check if the answer is dimensionally or logically possible. This prevents option-led guessing.

Functions: advanced-style trap

Problem: A multi-condition Functions problem often has an extra restriction hidden in domain, sign, interval or geometry.

Method: solve algebraically first, then filter using the domain/interval/geometry condition. In JEE Advanced, most wrong options come from skipping this final filtering step.

5. Common Mistakes & Traps

Finding inverse without restricting domain.
Treating every relation as a function.
Ignoring excluded values after algebraic simplification.

6. JEE Main Specific Strategy

For JEE Main, aim for fast recognition and clean substitution. Finish the first pass of Sets, Relations and Functions questions in 60–90 seconds each. Prioritise standard formulas, short sign/domain checks and option elimination only after the setup is correct.

7. JEE Advanced Specific Strategy

For JEE Advanced, expect combined conditions, hidden domains, multi-correct traps and integer-style answers. Build the solution from definitions, not memorised tricks. When a parameter appears, solve the general case and then filter using restrictions.

8. Quick Revision Summary

Check domain first.
Use horizontal-line logic for one-one.
For inverse, swap x and y only after confirming bijection.

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Finished this topic?

Keep the practice loop moving

Move straight from chapter-wise questions into a subject test, then loop back into weaker areas instead of ending the session here.

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